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^ffigmfj-fii^tispi  gJTnrrri 


L0GAR1THM0TECHNIA: 

OR,    THE 

^aWngof  lumbers 

CALLED 

LOGARITHMS 

To  Twenty  five Places, 

F  R  O  M    A 

GEOMETRICAL 

FIGURE, 

With  Speed,  Eafe  and  Certainty. 


The  like  not  hitherto  Publifhed. 


By  EVCLID  S P  EI D E L  L,  Philomath.    \ 


LONDON, 

Printed  by  Henry  Clark  for  the  Author,  and  are  to  bej 
Sold  by  Philip  Lea  ;  at  the  Atlas  and  Hercules  in  Cheap* fide, 
near  Friday-ftreet,  MDCLXXXVIII. 


Epjile  Dedicatory. 

Honours  Management;  this 
enfuing  Treatife  5  being  the 
Product  of  fome  leifure  Hours 
from  that  "Employ  ment9I  hold, 
under  your  Honours,  is  with 
^11  Duty  and  Subrniffion , 
humbly  Dedicated,  by 

r  M  4   Tour  Honours 


Aloft  Faithful  and 
Obedient  Servant , 


EUCLID  .SP.EIDE.LL. 


To- the  READER. 

.  : 

HAving  for  forne  Tears paj}  fiewnto  feveral  Perfons  the  Praxis 
of  the  following  Treatife,  and  alfo  communicated  to  them 
fomewhat  of  the  Dotlrin  leading  thereunto,  1  was  often  de fired  not 
to,  let .  them  fteep  in  oblivion,  but  to  publijh  the  fame,  whkh  was 
firft  promoted  by  my  honoured  Friend  Mr.  Peter  Hoot  Merchant, 
and  fecondedby  my  loving  Friend  Mr.  Reeve  Williams,  or  elfe 
they  had  not  feen  the  Fublique ;  what  1  have  done  therein  I  defire 
thee  to  take  in  good  part,  being  alfo  at  proportional  charges  my 
felf,  be  fides  my  compofing  thereof  to  make  it  communicable  to  theef 
rather  than  fuch  an  eafie  and  certain  way  to  make  Logarithm  Num- 
bers ( to  fo  many  Places  )  fliould  not  be  known  in  our  Native  TonguL 
J  have  called  them  Geometrical  Logarithms,  for  that  the  fir  ft  Inven- 
tors of  thofe  Numbers  had  not  adapted  Geometrical  Figures  to  them. 
But  the  Scheme  hereunto  annexed  having  fuch  Properties  and  Affec- 
tions as  Logarithm-  Numbers  have,  hath  made  me  fo  ftyle  them. 
What  I  have  done  herein  is  to  gratifie,  fuch  who,  have  a  Curiofity  to 
examine  Logarithm  Tables,  and  to  make  Logarithm  Number  to  fo 
fmall  Radiufes  as  are  fo  often  printed  for  common  ufes  with  brevity 
and  exaEtnefs.  Twojheets  of  the  Praxis  hereof  were  printed  fome 
time  before  the  reft,  which  having  found  kind  acceptance  with  di- 
vers, induced  me  alfo  tokt  the  Remainder  be  published:;  and  before 
the  printing  thereof  one  was  writing  upon  thofe  two  jheets,  and  was  fo 
fair  to  defire  my  con  fern  to  publijh  it,  which  I  readily  gave,  for  that 
J  knew  him  able  enough  to  do  it -y  and  when  to  be  m  Lei  fur  e  my  fclf 
to  attend  the  publishing  of  the  Refidue,  1  knew  not:  But  that  not  be- 
ing per for me d  by  hi ?n,  1  defire  thee  to  acceft  of  what  is  done  herein, 
as  time  and  leifure  hath  permitted.  1  (lull  not  need  to  write  how 
needful  Logarithm  Numbers  are  in  thofi  great  and  ufeful  Arts  of 
Navigation,  Aftronomy,  Dyalling,  Fortification  WGunnei 
Surveying,  Guaging,  Jntereit  and  Annuities,  &c,  When7  asth?\ 
are  fo  many  Bocks  written*andpublified  thereof  not  only  in  our  owA 
Language  but  in  many  others.  And  truly  the  fir  ft  Inventors  there  of\ 
are  not  a  little  to  be  had  in  reverence  for  making  and  perfecting  thofi\ 
Numbers  with  fo  much  Labour,  as  thofe  Methods  by  which  they  de- 
rived them  did  require,     Here  thou  may  ft    make  a  Logarithm  to 

7  <?r  8 


To  the  Reader. 

7  or  8  pUces  readily  and  eafily\but  to  25  places  would  have  been  very 
difficulty  if  not  impoffible^  for  the  fir  ft  Inventors  to  have  produced 
dfter  their  ways.  If  any  thing  herein  jhall  offer,  whereby  thou  may- 
oft  make  farther  Improvement^  Let  the  Publique  fhare  of  the  benefit 
thereof  Thus  w^ing  thee  good Succefs  in  all  thy  Studies^is  moft 
farnejily  defired  byy 

London, 
M<treb,t6.  1688. 

E.  Speidell. 


Advertifement  of  Characters,  or  Symbols,  ufed  in  this 

Treatife. 

-j-  1  -2  [More,  or  Addition, 

*  i  A  j  Multiplication, 

. — >  [  g^Lefs,  or  Subftraftion, 

=  j  US  (.Equal. 

ADVERTISEMENTS. 

ARTS  and  Sciences  Mathematical,  taught  in  Englifir 
or  French^  by  Mr.  Reeve  Williams  in  St.  Michael's 
Ally,  Corn-Hill^  at  the  Virginia  Coffec-houfe,  where  you 
may  have  thefe  Books. 

All  Sort  of  Globes,  Books,  Maps  and  Mathematical  Inflru- 
ments,  Made  and  Sold  by  Philip  Lea  Globe-maker,  at  the  Atlas 
and  Hercules  in  Cheap- fide,  next  to  the  Corner  of  Friday. ftYetu 


Gcome- 


( I ) 


Geometrical  Logarithms. 


chap     I, 

Bout  the  Year  167 4,  being  in  Company  with  Michael 
t  Dairy ,  a  Citizen  of  London  (  who  had  for  moll  part 
of  his  life  time  Eddi&ed  himfelf  to  Mathematical 
k*  Studies,  and  hath  publifhed  divers  practical  Pieces  of 
feveral  Parts  of  the  Mathematicks,  of  good  life  and  Delight ) 
and  difcourfing  about  making  Hyperbolical  Logarithms,  I  de- 
fired  him  to  give  me  a  Rule  to  make  the  Hyperbolical  Loga- 
rithm of  10,  from  the  Confideration  of  an  Hyperbola  infcri- 
bed  within  a  Right  Angled  Cone,  who  gave  me  this  Rule  fol- 
lowing. 

To  the  Number  propofed,  viz.,  10,  add  an  Unite,  and  fub- 
ftraft  from  it  an  Unite,  and  there  will  be  a  refult  of  ft  :  Then 
divide  1,  or  ioogooooo,  &c.  by-?,,  whichis  818181818,  &c. 
which  Cube  in  infinitum  ,  and  divide  every  one  of  them  (which 
will  be  a  Rank  of  Proportional   Numbers  )   by  the  proper  In- 
dices of  their  refpe&ive  Powers  ?  that  is  to  fay  by  3,  5,7,9* 
u,eh;.  Then  the  Addition  of  all  thofe  Quotes  will  make  the 
Logarithm  of  10. 
Finding  then  that  1  o  divided  by  U  ,msketh  8 1 8 1 8 1 8 1 8 1 8 1 8 1 , 
&c.  and  to  Cube  it  in  infinitum^   was  very  difficult,  I  rejefted 
the  Rule,  and  thought  it  then  not  much  more  eafle  than  "Brigg% 
way  :  Neither  did  he  tell  any  Reafon  or  Demonftration"  for  the 
faid  Rule  ;  and  becaufe  in  this  Example,  1  found  it  fo  intricai\ 
I  did  not  much  care  to  profecuteiit,  butneglefted  it.    Notion^ 
after  he  departed  this  Life;  and  fince  his  Death  reaming  the^| 
faid  thing,  and  trying  if  it  were  fervicfeablein  any  other  part  of" 
the  Hyperbola,  I  foon  found  it  a  Jewel,  acd  could  make  the 
Hyperbolical  Logarithm  of  10  at  twice,  that  is  to  fay,  from 
two  parts  numbered  in  an  Afymptote,  whofe  Fad  is  10,  with 
eafe,  certainty  and  delight,' and  have  made  the  Hyperbolical 

B  Logarithm 


(    2    ) 

Logarithm  of  2  to  25  places,  in  order  to  fee  if  the  learned  and 
laborious  Henry  Brings  Logarithms  were  true  to  15  places, 
which  were  made  arter  a  moft  laborious  and  difficult  way  of 
Extracting  Squat  e  Roots,  and,  as  I  have  heard,  was  the  work 
of  eight  Perfons  a  whole  Year,  and  that  without  any  proof,  but 
only  if  any  two  or  more  agreed  in  their  Extractions,  Line  by 
Line,  Step  by  Step,  it  was  taken  de  bene  ejfe,  which  was  a  work 
of  very  great  pains  and  uncertainty  :  However,  they  did  effedt 
it,  and  I  do  find  they  made  the  Logarithm  of  2  to  15  places 
very  true,  as  by  my-Operation,  hereafter  following,  will  ap- 
pear, being  done  to  25  Places,  and  afterwards  from  thefe  Hy- 
perbolical Logarithm  deduced  Brigg's  Logarithms  \  both  which 
Figurative  Operations  were  performed  and  examined  by  me  in 
8  Hours  time.  I  took  this  pains  to  make  the  Hyperbolical  Lo- 
garithm to  2$  Places  \  in  order  alfo,  to  fee  if  the  moft  ingeni- 
ous and  laborious  James  Gregory's  Hyperbolical  Logarithm 
would  agree  with  this  of  mine,  which  he  hath  in  his  Qga- 
dratnra  Circuit  &  HyperbeUj  Printed  at  Padua  ;  but  I  find  Mat 
his  Logarithm  of  2  correfponds  with  mine  but  to  17  Places,  I 
mull  confeOj,  I  did  not  take  the  Pains  to  raife  the  Logarithm 
of  2  to  25  Places,  according  to  the  Do&rine  he  hath  delive- 
red in  that  moft  Learned  Piece,  but  am  contented  that  this  ea- 
fie  and  certain  way  I  deliver  here,  and  by  the  Operation  there- 
of the  Hyperbolical  Logarithm  of  2  to  25  Places,  is  as  true  in 
the  iail  as  in  any  where,  and  may  be  examined  in  a  few  Hours  ; 
fo  that  any  Body,  if  he  pleafe,  may  be  hi$  own  Examiner  and 
Judge,  if  this  Way  be  not  eafie,  certain  and  fpeedy. 

Having  made  feveral  Logarithms4br  Digit  Numbers,   and  ' 
Mixt  Numbers,  as  for  i|,  ij,  which  are  hereafter  inferred. 
The  Rule  delivered  by  Michael  Dairy y  is  of  admirable  nfe  and 
benefit  in  Squaring  the  Hyperbola,   and  making  Logarithms 
from  it.  \ 

Some  time  fince  the  death  of  the  faid  Michael  Dairy ,  1  Hie  wed 
^into  Mr.  John  Collms^  whom  1  knew  had  been  a  great  Familiar 
and  Friend  to  the  faid  Michael  Dairy,  the  Figurative  Work  of 
my  making,  the  Hyperbolical  Logarithms,  according  to  the 
faid  Dairfs  Rule,  who  feerned  very  well  pleafed  with  it,  ac- 
knowledging it  to  be  the  fpeedieft  way  could  poffibly  be,  of 
Squaring  the  Hyperbola,  and  making  the  Logarithms  from  it, 
and  after  a  little  paufing  upon  it/replyed,   That  Dairy  muft 

have 


( I ) 

have  this  Rule  out  of  the  faid  James  Gregory**  Works.  I 
made  anfwer,  Not  from  his  faid  Qmdratnra  Circuit  &  Hyper- 
bola: He  anfwered,  No,  from  \\i%Exercitationes  Geometric*, 
printed  at  London,  i568,  a  Book  I  had  not  feen  nor  heard  of 
till  then*:  And  as  the  faid,Mr.  Collins  had  been  always  very  frank 
and  free  to  communicate  any  Mathematical  thing  tome,  fo  I 
held  my  felf  obliged  to  acquaint  him  firft  with  this  Work.  He 
feemed  to  admire,  That  Michael  Dairy  fhould  keep  fuch  a  thing 
from  him,  who  had  been  fo  great  a  Familiar  with  him  in  thefe 
Studies.  Not  long  after  my  difcovery  hereof  to  the  faid 
Mr.  Collins,  he  alfo  departed  this  Life ;  whofe  Death,  all  that 
were  Mathematical,  and  knew  him,  lamented  riot  a  little:  For 
he  was  not  only  excellent  in  Mathematical  Arts  and  Sciences, 
but  of  a  very  good,  affable  and  frank  Nature  to  Communicate 
any  thing  he  knew  to  any  Lover  and  Enquirer  of  thofe  things, 
and  hath  left  behind  him  thofe  Mathematical  Works  which  will 
continue  his  Fame  amongft  the  Lovers  and  Students  therein. 
He  alfo  in  his  Life  time,  promoted  the  Publifhing  of  other 
Men's  Mathematical  Works  ;  as  the  Elaborate  Algebra  of  the 
Learned  JohnKerfy,  who  was  my  Father's  Difciple  about  \6^r : 
And  alfo  of  the  Learned  Baker* %  Algebra,  and  feveral  others. 
He  was  a  Man  of  great  correfpondence  with  Mathematical  Per- 
fons  in  foreign  Parts,  and  thereby  could  give  Information  of 
any  N?w  or  Old  Mathematical  Book  ;  and  till  my  Acquaintance 
with  him,  I  was  ignorant  of  Foreign  Authors ;  being  but  young 
when  my  Father  dyed,  and  not  then  having  taken  any  Pains  in 
thefe  Studies :  So  that  by  the  faid  Collins  %  Information  and 
Means,  I  have  heard  of,  and  feen,  fome  Foreign  Mathematical 
Authors  of  Note  and  Efteem. 

After  the  faid  Mr.  Collins  had  told  me  of  James  Gregorys  faid 
Exercitationes  Geometric*,  fold  by  Mofes  Pitts  in  Paul's  Church  - 
Yard,  I  bought  there  one  of  them  -7  and  do  find  that  Michael 
Dairy  had  deduced  this  Rule  from  the  faid  Book  :  Wherein  t!K 
faid  James  Gregory  hath  made  the  Squaring  of  the  Hyperbola,! 
sn  Exercife  Geometrically  dernonftradng  the  Quadrature  of  the  % 
Hyperbola,  fome  time  before  published  by  the  Induflrious  and  f 
Lucky  Nicholas  Mercator  ;  whoby  thehappy  difcovery  of  fome- 
Properties  in  the  Hyperbola,  hath  made  all  the  Ways  of  Squa- 
ring the  Hyperbola  flowing  from  the  fame,  very  eaii'e,  certain 
and  delightful :  And  became  neither  of  them  have  exemplified 

B  2  their 


(4) 

their  Dodrinc  and  Rules  with  Figurative  Work,  fo  large  as  to 
25  Places.  I  have  here,  to  llluftrate  their  Admirable  Works, 
inferted  divers  Figurative  Operations,  whereby  the  Reader  and 
Student  may  fee,  and  have  that  Satisfadion  in  Fad  and  Opera- 
tion, which  is  fo  pleafing  and  defirable  by  every  one. 

I  (hall  not  here  trouble  the  Reader  with  any  Sedions  of  the 
Cone,  whereby  he  may  fee  the  rife  and  geniture  of  an  Hyper- 
bola, from  that  Body,  but  content  my  felftolhew  him  from  a 
Square  and  an  infinite  company  of  Oblongs  on  a  Superficies,  each 
Equal  to  that  Square,  how  a  Curve  is  begotten  which  fhail  have 
the  fame  properties  and  affedions  of  an  Hyperbola  infcribed 
within  a  Right  Angled  Cone  :  And  feeing  a  Curve  made  after 
this  manner  following,  doth  become  fuch  an  Hyperbola,  the 
Dodrines  and  Analogies  delivered  and  difcovered  by  thofe  two 
Ingenious  Artifts  Mercator  and  Gregory^  may  be  applyedto  this 
Curve  fo  often  as  need  aind  occafion  doth  require. 

And  not  to  detain  the  Reader  any  longer  from  knowing 
how  to  make  this  Curve,  we  proceed  to  defcribe  the  fame  ac- 
cordingly. 

There  is  a  Square  ABCD,  whofe  Side  or  Root  is  1  o,  let  DP> 
be  prolonged  in  infinitum^  and  continually  divided  equally  by 
the  Root,  or  DB ,  and  thofe  Equal  Divifions  numbered  by  1  o ,20, 
30,40,50,60,70,  &c.\xiinfinitHm:  Upon  thefe  Numbers  let 
Perpendiculars  be  ereded,  which  call  Ordinates,  and  each  of 
thofe  Perpendiculars  of  that  length,  that  Perpendiculars  let  fall 
from  the  aforefaid  Perpendiculars  to  the  Side  orBafe  CD(which 
call  Complement  Ordinates )  the  Oblongs  made  of  the  Ordi- 
nate Perpendiculars,  and  Complement  Ordinate  Perpendiculars 
may  be  ever  Equal  to  the  Square  AD,  which  is  eafily  done  thus, 
forit  is  ?-;;  'fo  l°/o  'fo  &c.  produces  the  Length  of  the 
Ordinate  Perpendiculars ;  for  100  divided  by  20  maketh  5  for 
the  Length  of  the  Ordinate  Perpendicular  20  E.  And  100 
divided  by  3o,giveth  3333333)^.  for  the  Ordinate  Perpendi- 
cular 30  F.  And  ioq  divided  by  4©  produceth  >5  for  the  Or- 
/dinate4oG,  and  foof  the  reft.  And  Geometrically  it  is  as 
20  D  is  to  BD,  fo  is  AB  to  AH,  equal  to  20  E,  as  before,  for 
that  the  Angle  ACH  is  equal  to  C20  D,  and  fo  of  the  reft. 
And  for  the  Length  of  the  next  Ordinate,  you  fay,  as  30 
D  to  BD;  fo  AB  to  AK,  which  is  alfo  Equal  to  30  F.  And  for  the 
Ordinate  40  G,  fay,as  40  G  to  BD,  fo  AB  to  AM,  which  will  be 

Equal 


(5  ) 
equal  to  40  G,  and  fo  of  all  the  reft,  whereby  you  have  all  the 
Perpendiculars  upon  the  prolonged.  SideDB^both  Geometrically 
and  Arithmetically  ;  the  fame  Propoi  tion  Is  to  be  obferved  for 
any  Intermediate  Parts. 

Now,  for  all  the  Perpendiculars  which  are  let  fall  from  the 
aforefaid  Perpendiculars  or  Ordinates  to  the  Bafe  CD,  which 
call  Complement  Ordinates,   the  Geometrical  Proportion  for 
NE,equal  D  20  is  as  HA  to  AC,fo  CD  to  C^U©  equal  to  NE,and 
for  the  Complement  Ordinate  OF  equal  D  30,  it  is  as  KA  to 
AC,    foCDtoD3o  equal  OF,  and  fo  of  the  reft.     Now.  for 
NE  Arithmetically,   fay,  as  5  to  10,  fo  10  to  20  equal  to  NE, 
equal  to  D20  ;  and  for  OF,  fay,  3533333333  to  10,  fo  10  to 
30  equal  OF,  which  is  equal  to  (££30,  and  for  PG  equal  to  D 
40,  fay,  as  >5  is  to  10,  fo  10  to  40  equal  to  PG,  equal  to  D  40; 
and  fo  for  all  the  reft  of  the  Complement  Ordinates  Handing 
upon  the  Bafe  CD,  whereby  it  doth  appear,  That  all  the  Ob- 
longs made,  of  the  Ordinates,  and  Complement  Ordinates  are 
each  of  them  equal  to  the  Square  AD,  which  is  here  100;  for 
the  Oblong  ED  being  made  of  E  20  and  D  20,    is  by  the  13  of 
of  the  6  Euclide  equal  to  the  Square  AD,  for  Q_  20  is  a  Mean 
Proportional  between  D20,  and  20  R,  and  Q^  20  is  found  to  be 
equal  AB,  fo  is  the  Oblong  or  Parallelogram  ED  equal  to  the 
Square  AD,  and  the  like  Demonftration  ferves  for  all  the  Ob- 
longs or  Parallelograms  (landing  upon  the  Bafe  CD,  by  the 
Tips  orAngularPoints  of  thofeParallelograms,or  from  the  Ends 
of  all  the  Ordinates  (landing  upon  20,30,40,  50,60,70,  in 
infinitumy&xaw  theCurve  Line  from  A  towards  E,fo  {hall  you  de- 
fcribe  the  Curve  AEFGS,which  Curve  you  fee  is  begotten  with- 
out any  conlideration  or  refped  to  the  Se&ion  of  a  Cone,  and 
yet  becomes  the  fame  in  all  reipe&s,  to  have  the  fame  Affecti- 
ons and  Properties  of  an  Hyperbola  derived  from  the  interfe- 
ring of  a  Right  Angled  Cone,  as  (hall  be  (hewed  in  the  next 
Chapter.  . 

You  may  obferve  the  Complement  Ordinate  NE,  being  e&jal 
toD20,  is  equal  to  twice  Radius.  And  if  CD  be  made  the  Ra- 
dius of  a  Circle,  then  is  NE  equal  to  D  20,  equal  to  the  Tan- 
gent of  twice  Radius,  for  D  20  becometh  the  Tangent  of  twicrt 
Radius.  Alio  it  is  mahifeft  that  the  Complement  of  the  Tan  1 
gent  equal  to  twice  Radius,  is  aifo  equal  to  half  the  Rsdius  7$ 
That  is,  the  Tangent  Complement  of  D  20,  is  20  E  equal  to  5. 

And 


CO 

And  feeing  the  Radius  is  ever  a  mean  Proportion  between  the 
Tangent  and  the  Tangent  Complement,  therefore  each  Oblong 
is  equal  to  the  Square  AD.  ..    . 


c  h   a   p.     jr.. 

N  the  former  Chapter*  we  have  fhe wed  the  begetting  of  a 
Curve,  without  any  regard  to  the  Section  of  any  Solid  Bo- 
dy ;  and  now  it  remaineth  to  prove  that  this  Curve  hath  the 
fame  Properties  and  AfTedions  that  an  Hyperbola,deduced  from 
the  Section  of  a  Right  Angled  Cone. 

I  remember  fome  time  before  the  death  of  John  Ccllins^  he 
told  mejt  was  a  great  Wctk  of  the  Learned  Vincent  or  Mag- 
nan,  to  prove  that  Distances  reckoned  in  the  Afymptote  of  an 
Hyperbola,  in  a  Geometrical  Progreffion,  and  the  Spaces  that 
the  Perpendiculars,  thereon  ere&ed,  made  in  the  Hyperbola, 
were  equal  the  one  to  the  other.  This  Property  is  now  very 
well  known,  £he  Hyperbola  hath,  and  this  Curve  hath  the  fame 
Property  ,  which  is  difcernable  altnoft  intuitu.  In  the  Hyper- 
bola, they  call  the  Prolonged  Line  DB  in  infimtnm1  from  the 
Point  B,  an  Afymptote.  And  here  in  this  prolonged  Line  from 
B,  on  20,40,  80,160,320/640,  eh; .  let  the  Ordinate*  touch 
the  Curve  in  EFGS,  &c  f  fay,  That'thofeTrapezias  with  the 
Curve  Line  (  or  Hyperbolical  Spaces  )  are  all  equal  the  one  to 
the  other.  In  the  Right  Lined  Trapezias  thereon,  it  is  mani- 
fell  they  are  ail  equal  the  one  to  the  other  by  feveral  Propositi- 
ons of  the  6th  Book  of  Euclid :  For  in  the  Right  Lined  Trape- 
zia 2EAB  the  fide  AB  is  twice  EZ,  and  by  the  former  Chap- 
ter it  was  found  that  GY  is  half  EZ,  by  faying,  As  AB  to  EZ, 
fo  EX  to  GY.  And  the  Right  Lined  Trapezia  ZEAB  (hall  be 
th^veforeequalt0  75  :  Now,  forafmuch,  as  in  the  Right  Lined 
Trapezia  YGEZtheBafe  of  thatYZ,  is  double'  to  ZB,  but 
the  Perpendiculars  are  in  the  Ratio  of  AB  to  EZ  ;  for  as  before, 
if  is  as  AB  :  EZ ::  EZ :  GY,  therefore  the  Right  Lined  Trape- 
Yia  YGEZ  equal  to  the  Right  Lined  Trapezia  ZEAB,  and  fo 
will  all  Right  Lined  Trapezia's,  fo  Bafed  and  Perpendicular'd, 
be  equal  the  one  to  the  other.     The  Trapezia  LYEZ  is  equal 

to 


(7  ) 

to  the  Square  AD,  becsufe  ZY  *  ZE  is  equal  toAB  *  AB,  as 
in  the  foregoing,  Chapter,  the  Oblong  G2  is  half  the  Parallelo- 
gram LZ,and  the  Triangle  GEI  half  the  Parallelogram  LI.  Now 
the  Parallelogram  GZ-|  the  Triangle  GEI  (half  the  Paralle- 
logram LI  )  is  equal  to  the  Right  Lined  Trapezia  ZEAB,  for 
in  Numbers 20  *  2*5  ==150  -j~  half  50  equal  to  75. 

Thus  you  fee  the  Right  lined  Trapezias  cumbred  upon  the 
prolonged  fide  in  a  Geometrical  Proportion  are  equal  the  one 
to  .the  other  j  it  remaineth  now  to  prove  the  mixed  Trapezias, 
that  is,  the  Trapezias  (landing  upon  the  fame  bafis  but  joyned 
aloft  with  this  Curve  are  alio  equal  the  one  to  the  other. 

Firft  let  it  be  obferved,  That  thefe  Curvilihed  Trapezias 
(  or  Hyperbolical  Spaces )  are  ever  lefs  than  the  Right  lined 
Trapezias,  becaufe  all  the  Points  in  the  Curvilined  Trapezias 
fall  within  the  Right  line  that  joyns  the  Right  lined  Tra- 
pezias :  And  is  thus  proved  in  the  Right  Lined  Trapezia 
BZEAi  let  there  be  in  the  BafeZ  B  upon  the  point  5  Eredted 
a  Perpendicular  to  touch  E  A  in  T  then  is  T  5  equal  to  A  B 
lefs*  T  which  is  half  Q^E,  that  is,  10  lefs  2,  5  equal 7,  5 
=  T  5.  But  by  the  foregoing  Chapter,  if  a  Perpendicular  be 
ere&ed  upon  the  faid  Point  5  to  V,  (  to  touch  the  Curve  in  V  ) 
fo  that  the  Parallelogram  VD  (hall  be  equal  to  AD,  as  in  the 
former  Chapter,  then  will  it  be  AD  divided  by  D5  =  5  V, 
which  is  100  divided  by  15,  produceth  6  666666  for  the  true 
Length  of  5  V,  whereas  before  5  T  is  7,  5.  By  the  fa'trfe 
means  may  all  the  intermediate  Points  in  this  Curve  Line  EVA 
be  found  to  fall  within  the  Right  Line  AE,  that  is,-  between  the 
Line  EA  and  ZB,  and  therefore  the  Right  Lined  Trapezia 
ZE  TAB  greater  than  the  Curv-ilined  Trapezia  (  or  Hyperboli- 
cal Space)  ZEVAB. 

Now,  forafmuchas  we  have  proved  that  the  aforefaid  Right 
Lined  Trapezias  are  ever  equal  the  one  to  the  other,  it«ftill 
now  follow,  That  feeing  the  Curve  palling  by  all  thofe  Poh'js 
which  are  Extremities  of  the  Right  Lined  Trapezia.,  ("as  well  as' 
the  Curvilined  'Spaces,  being  upon  thj  fame    Bafes  always ) 
and  this  Curve  being  generated  continually  by  one  and  the  fjitti 
Ratio,  as  in  the  former  Chapter.     That  therefore  "he  Curvi- \ 
lined  Trapezias  Handing  upon  Geometrical  Pi  oportional  Bafes,  * 
fliall  be  alio  equal  the  one  to  the  other,   which  is  the  Affe&ion 
and  Property  of  the-Hyperbola.     And  fo  the  Doctrines  and 

Precepts 


(  8  ) 

Precepts  delivered  by  thofetwo  Famous  Geometers,  Mercator 
and  Gregory,  for  the  Squaring  of  the#  Hyperbola,  be  applyed 
to  this  Geometrical  Curvilined  Figure  }  and  from  it  derived 
Logarithms,  which  may  be  called  Hyperbolical  Logarithms. 

The  way  and  means  to  find  the  Hyperbolical  Spaces  in  Num- 
bers, (hall  be  (hewed  in  the  following  Chapters*  .      , 


CHAP.       III. 

IN  this  Chapter  we  will  confider  that  moft  admirable  difco- 
very  (I  fuppofe Mercator  made)  upon  drawing  the  Dia- 
gonal CB,  which  by  conllru&ion  cutteth  all  the  Perpendiculars 
ftanding  upon  the  Bafe  CD  at  Equal  Angles,  and  in  fuch  Di- 
ftances  from  the  Bafe  CD,  as  doth  unravel  the  Myftery  of  his 
infinite  Series,  and  make^  the  Quadrature  of  the  Hyperbola 
more  eafie  arid  certain  than  any  1  ever  faw  or  heard  of. 

The  Diagonal  CB  being  drawn  doth  give  the  firft  Term  of 
a  Geometrical  Progreffion  or  Infinite  Series  between  10  and  20, 
01-30,40,50,60,70,80,905  &c. 

That  is  to  fay,  Would  you  know  the  firft  Term  of  an  Infinite 
Series  (  or  Numbers  Geometrical  Proportional  continued  }  be* 
tween  10  and  20,  the  fumm  of  all  which  fhall  be  juft  20.  Ha- 
ving from  Z  drawn  the  Line  ZC,  to  cut  BA  in  H,  which  taken 
off,  and  applyed  to  CD,  from  C  to  N  eqnal  NB,  becaufe  the 
Angle  BCNT  is  equal  to  the  Angle  CBN :  1  fay  that  NB  is  the  firft 
Term  of  an  Infinite  Series  between  CD  equal  AC,  and  the  Per- 
pendicular NE  equal  DZ,  which  may  be  done  by  Squaring  AC, 
and  dividing  it  by  the  Side  or  Number  given,  the  Complement 
whereof  to  10  is  the  firft  Mean  or  Term  of  that  Infinite  Series, 
ft£tiall  the  firft  Term  of  the  Infinite  Series  between  10  and  20 
be  found  5  :  Thus  in  Numbers,  10  *  10  =  109  %l'=  5,  the 
Complement  whereof  to  10  is  5,  equal  CN,  equal  NB  for  the 
firft  Term  of  an  Infinite  Series  between  to  and  20,  whofe  fumm 


is  20 


(9) 

is20,   as  by  the  Arithmetical  work  in  the  Margent^  where 
a,bjC)d,eif)  &c.  are  a  Rank  of  Geo- 
metrical Progreffional  Numbers,    whofe 
infinite  Sum  would  make  but  20,  and  is 
demonftrated  by  the  7  >£<#  *W 

And  in  Numbers  thus.  As  10  lefs  5  is 
to  10  what  10?  the  Quotient  will  be 
found  20  for  the  whole  Summ  of  that  in- 
finite Series  between  10  and  20  whofe 
fir  ft  Term  is  5 

In  like  manner  would  you  know  the 
firft  Term  of  an  infinite  Series  between 
10  and  30  Divide  the  Square  of  ac  =  100 
by  3  o  the  Quotient  will  be  found  3333333 
whofe  Complement  to  10  is  6666666 
I  fay*  That  6666666  is  the  firft  Term  of 
an  Infinite  Series  between  10  and  30, 
as  by  the  Arithmetical  Operation  in 
the  Margent ;  And  briefly  thus,  as 
10:  :$3333$3=66$6666:  10::  10:30  ft) 
is  30  the  whole  Summ  of  all  thofe  infinite 
Progreflional  Numbers  between  10  and 
30.  In  the  figure  you  draw  the  line  $C 
which  cutteth  BA  in  K,  I  fay  that  BK 
transferd  from  C  toO  equal  Or  is  the 
firft  Term  of  an  infinite  Series  between 
AC  and  OF  equal  D*.  And  PS  be  the  firft  Term  equal  7, 5 
equal  PC  between  AC  and  PG  equal  DY  =  40  between  10 
and  40,  for  as  before  10  — 7,  5  =2,  5 :  10  ::  10=40  fois 
40  the  whole  Summ  of  an  infinite  Series  between  10  and  40* 
whofe  firft  Term  is  7,  5  and  fo  of  all  the  refh  On  this  great 
Myftery  depends  much  thefollowing  fquaring  of  the  Hyperbo- 
la and  hath  made  it  fo  intelligible  andeafie. 

Hence  you  may  Note,  That  if  you  would  know  the  £)$ 
mean  of  an  infinitrr^ries  between  any  other  Number,  and  that 
Number  doubled,  tripled ,•  quadrupled,  &c.  you  may  from  this 
Root  10  deduce  it,  as  for  Example,  let  CD  be  12  and  I  would 
know  the  firft  Term  of  an  infinite  Series  between  12  and  24  \ 
the  double  of  12 1  fay  as  10  is  to  5  what  12  facit  6.  for  fuppo-  J 
fing  CD  =  10  it  was  before  found  5.    Therefore  between  12 

C  and 


a. 

IO 

t 

s 

c. 

2.5 

d. 

1.25 

e. 

.  .  625 

/• 

.  .  3125 

h. 

. .  15625 

i. 

.  .  .78125 

k. 

.  . .  390625 

1. 

•  •  •  1953125 

rn. 

. .  ..  97655^5 

19 

.  99.  0234375 

a. 

10. 

b. 

.  6. 666666666 

c. 

.  4. 444444444 

d. 

. 2.962962962 

e. 

•  1. 975308641 

f 

.  1.  316872427 

£' 

.  ..8779H951 

b. 

.  ..585276634 

i. 

,  .  .390184422 

ft. 

k 
d 

I'- 


ll 

6 
1 
U 


5 

75 

375 

1875 
*375 
4<J875 
134375 
1 17 1 87 5 

•  5855375 


f 


(10) 

and  24  you  fay  10  :  5  ::  12 :  —6  for  the  firft  mean  of  an  In* 
finite  Geometrical  Progreflion  between  12  and  24.  And  is 
thus  by  the  former  Analogy  proved  by  faying  as 
12 — 6=6  :  12:  :  12  :  =  24  fo  is  24  found  to  be  the  total 
Summ  of  an  infinite  Series  between  12  and  24  as  by  the  Opera- 
tion in  the  Margent  will  appear. 

Alfo  if  it  were  required  to  find  the  firft 
mean  between  12  and  three  times  that 
number  ^£.36.  Say  as  10:  6666666:  12::  8. 
And  fo  by  Tabulating  or  working  this 
Eximple  as  you  do  the  former  the  total 
Summ  of  that  infinite  Series  between  12 
and 36  (the  firft  Term  being  found  8) 
will  amount  to  36 :  And  for  proof  you  fay 
12-8=4:  1.2  : :  12 :  =  36  fo  is  36  the 
whole  Summ  of  that  infinite  Series  or  Geo- 
metrical Progreflion  between  12  and  36, 
the  firft  Term  being  8  as  was  defired. 

If  it  (hall  be  required  to  know  an  infinite 
Series  between  10  and  any  other  Numbers 
to  know  the  firft  Terni  between  10  and  15 
that  is,between  DC  and  D5,  draw  5C,  and 
it  cutteth  BA  in  E,  I  fay  that  BE  is  the  firft  Term  of  an  infinite 
Series  between  1  o  and  1 5  as  by  the  Ope- 
ration in  the  Margent,  and  as  before  taught 
l{l  =  6666666  j  which  fubftraft  from  10 
leaveth3333333  for  the  firft  Term.  And 
by  the  former  Rule  is  proved  thus  \  As 
io-3333333=6<566566  :  10  :  r  10  : -==1 5. 
Thus  may  you  find  the  firft  of  any  Term  of 
an  infinite  Series  between  10  and  any  other 
Number. 

And  if  it  fhould  be  defired  to  know  the 
firft  Term  of  an  infinite  Series  between  any 
two  other  Numbers*. ^ is  for  Example,  I 
would  know  the  firft  Term  of  an  infinite 
Series  between  1 2  and  16,  to  do  it  Geome- 
trically you  muft  fuppofe  BA  —  ti«  And 
then  counting  16  from  D  in  the  line  DB 
prolonged  bythat  Point^and  C  draw  a  line  which  will  cut  the 

line. 


23v994H°<^5 


A. 

b. 
C. 

d. 


IO 
3333333333 
I  I  II  II  II  I  I 

37037O37O 

123456793 

•  41 1 52264 

•  I37I7421 

•  •  457*473* 

•  •  15H157 

.  .  .  508052 

•  •  *•  159350 
m.  .  l[   .  .  55450 


£■ 
h. 

i. 

I 


»-4j  99991*11 \ 


( m 

incBA  (tiowreprefentingi2  )  in  a  point,  which  taken  From 
B,  will  be  the  length  of  that  line  Geometrically,  and  Arithme- 
tically it  will  be  found  by  faying,  as  i<5  D  to  DC,  fo  i<5  B  =  4 
to  B  3  in  the  line  BA  numbredfrom  B  to  A  when  BA  is  12.  In 
Numbers  the  Proportion  ftands  this,  16 :  12 : :  4 :  =  3  which 
3  is  the  firft  Term  of  an  infinite  Series 
between  12  and  1 6  as  was  required,    a.  12 
and  is  manifeft  by  the  Operation  in    b.  .  3 
the   Margent,    which   to  prove  by    c.  .  .75 
the  foregoing  Rules    you    fay,    as    d.  .  .1875 
12 — 3=9  :  12  : :  12  :  —  16;  which  is    e.  .  ..  4587s 
to  fay,  as  9  to  1 2,  what  1 2  fieit  1 6  for    f.  .  . .  1 17 1 875 
the  whole  Summ  of  that  infinite  Series    g.  .  . ,  •25295875 

between  12  and  i5,  the  firft  Term    h 732421875 

being  found  as  before  to  fee  3.  i 183 10545875 

Thus  have  you  that  great  Myftery 

unfolded  of  finding  Geometrically  and     1 5,  9999389548437$ 
Arithraetrically  the  firft  Term  of  a 

Geometical  Progreffion  with  the  whole  Summ  of  that'  in- 
finite Progreffion  or  Series  between  any  two  Numbers,  which 
is  the  main  thing  I  conceive  that  famous  Mercator  was  fo 
lucky  in  difcovery  thereof,  and  doth  unravel  the  Myftery  of 
fquaring  the  Hyperbola,  as  will  be  manifeft  in  the  next  Chapter 
following. 


c  H  A  P.    IV. 

IObferVe  from  the  (aid  Learned  Gregory**  Exercitationes  Geo- 
Metric*,  he  givcth  three  Quantities  or  Spaces  contiguous  to 
the  Vertex  A, which  ftiall  be  all  equal  theone  to  the  other  which 
is  very  true  and  perfpicuous,  and  then  (hews  how  to  fincL|he 
Areas  of  them  fever  ally  as  in  page  9,  10,  11,  and  12  of  md 
Book. 

And  here  we  muftconfider  them  all  three  before  we  come 
to  understand  Dairfs  Rule,  which  is  but  a  Dedu&ion  from 
thefe,  as  will  appear  hereafter.  And  now  I  begin  tocoafider 
She  faid  three  feveral  Quantities  or  Spaces  all  contiguous  to  | 

G  2  the 


(   I  *  ) 

thcVertexhy  and  Qf  a  different  form,  and  yet  equal  the  one  to 
the  other. 

Let  it  therefore  now  be  fhewn  thofe  three  feveral  Spaces  dif- 
fering inform,and  yet  equal  the  one  to  the  other  contiguous  to 
the  Vertex  A,  which  (hall  reprefent  the  Curvilined  Trapezia?  or 
Hyperbolical  Space  for  a.  The  firft  Curvilined  Trapezia  or  Hy- 
perbolical Space  for  g  let  be  ZBAVE,  which  is  intelligible 
intuitu.  The  fecond  let  be  AVENC,  which  is  equal  to.  the  for- 
mer 2BAVE  by  the  43  of  the  1  of  Euclid,  becaufe  the  Paral- 
lelogram EZBH  (  BH  being  equal  to  2E  )  is  equal  to  the 
Parallelogram  HACN  (  HA  being  equal  to  HB.  )  Now  for  as 
much  as  the  GuryiJinedTriangle  AVEH  is  common  to  both  the 
faid  Curvilined  Trapezius  or  Hyperbolical  Spaces,  it  remain- 
eth  therefore,  that  thefe  two  Curvilined  Trapezius  or  Hyper- 
bolical Spaces  ZEVAB,  and  AVENC  are  equal  the  one  to  the 
other.  And  now  to  find  out  the  third  Curvilined  Space  con-' 
tiguous  to  the  Verte x  A,  and  yet  equal  to  either  of  the  other 
two,  but  differing  inform,  doth  require  a  little  further  con- 
fideration  which  from  him  is  directed  thus.  |  And  is  manifeft  by 
the  figure,  divide  BZ  in  two  equal  parts  in  5 .  Then  as  before 
taught  will  it  be  as  5  D  :  DC : :  5  B  :  BK=DX  make  CX  equal 
to  Cn,  or  to  find  DH  it  is  as  D  s  :  DZ : :  DC :  Dn  upon  n  ereft 
a  Perpendicular  to  touch  the  Return  or  Continuation  of  the 
Curve  on  the  other  fide  of  A,  from  A  towards  ®  in  2.  So  is 
this  Curvilined  Figure  or  Hyperbolical  Space  nsAVX  C  differ- 
ing in  form )  equal  to  either  of  the  other  two  ZEVAB  or 
C  AVEN.  Andfrom  finding  the  Area  of  this  Curvilined  Figure 
or  Hyperbolical  Space  HzAVXis  derived  Dairy's  Rule,  which 
is  but  a  Deduction  from  the  finding  of  the  ^ireas  of  the  other 
two,  as  will  hereafter  appear. 

Arithmetically  DX  is  found  by  faying,  as  15  to  10 
what  1  o  ?  facit  66666666 ,and  1  o  lefs  66666666  reft  33333333 
for  Cn  fo  is  Xn  equal  to  66666666^  pr  Dn  may  be  found 
thti*.;  as  D5  :  DZ: :  DC :  Dn,  which  in  Numbers  is,  As 
1^  2©  r:  10:—  1.  33333333  for  Dnv 

James  dregory  in  the  4  Propofition  page  10, 11  of  hhExer- 

citMiwes  G.e.ometrict  doth  contemplate  firft,  the    Second  of 

/thefe  three  Cutvilinedrr^^^or Hyperbolical  Spaces,  that 

(is  to  fay,    the   Curvilined  Trapezia  ox  Hyperbolical  Space 

GAVEN,  and  in  that  4  Propofition  after  and  long  alearned  De- 

monftration 


The  I  „fi,ite  Siries  of  Nnm-  The  Quotes  to  be  Added. 

btrs  Proportional. 


fi 

50, 0000000000000 

, 

*5o 

, ooooooooooooo  A 

B 

2S 

II 

-12 

5           B 

C 

12$ 

III 

*  4 

1666666666666   C 

D 

IV 

5625        D 

E 

3125 

V 

* 

625          E 

F 

1562$ 

VI 

' — 

2604: 66666666   F 

G 

78125 

VII 

* 

IU6O71428571  G 

H 

39062$ 

VIII 

— V 

48828125     H 

I 

15,53125 

IX 

* 

217013888888  I 

K 

9765625 

X 

• — 

9765625     K 

L 

48828125 

XI 

* 

44389204545  L 

M 

244140625 

XII 

— » 

20345052083  M 

N 

122070312 J 

XIII 

* 

939^014037  N 

O 

61035 '5625 

XIV 

— 

4359654^17  O 

P 

305i757«l2J 

XV 

* 

2034505206  P 

Q. 

15258789062 

XVI 

— 

953674316  Q, 

R 

7629394531 

XVII 

* 

448787914  R 

S 

381469726s 

XVIII 

— 

21 1927625  s 

T 

1907348632 

XIX 

* 

100386770  -|' 

V 

953674316 

XX 

— ■ 

47683715 

W 

476837  is8 

XXI 

* 

22706531 

X 

238418579 

XXII 

— 

10^37203 

Y 

1 19209289 

XXIII 

* 

5183013 

Z 

59604644 

XXIV 

— 

2483527 

29802322 

XXV 

* 

t 192093 

14901161 

XXVI 

— 

'  573122 

7450580 

XXVII 

* 

283355 

3725290 

XXVIII 

— - 

133C46 

1 8626+5 

XXIX 

* 

64229 

931322 

XXX 

— 

31044 

465661 

XsXl 

* 

1502: 

232831 

XXXII 

— » 

7276 

116415 

XXXIII 

* 

3527 

58207 

XXXIV 

— . 

1712 

29103 

XXXV 

* 

83 1 

I45SI 

XXXVI 

r* 

404 

7275 

196 

3637 

XXXVIII 

— 

S>6 

1818 

XXXIX 

* 

46 

909 

XL 

— 

22 

454 

XL1 

*■ 

II 

277 

XLII 

_ 

5- 

2 

I 

10,  00000000  == 
The  whole  Sumra 

Logarithm? 
of  2.   i 

693 147 1 805 59945 

A— B:A::Ar= 

....   Halfth 

eLo-7 

6,  66666666  = 

A— C :  A  :  :  A  = 

Imparet 

garithir 

5493O6144334055 

Logarithm  1 

IO98612288668H© 

3,333333333  = 
Pares 

Logarithm  -j 
of  the  differ.  / 

B— D:B:t  B:  = 

n2&> 
Log-  3 

,(.0546510810816$ 

A 

-B.C-D.  E-F.G~t 
=  3>  333333333 

I       '  of°«.  ! 

143841036225890 

A- 

-B-l-C— D+E— F-|- 
=  3,333333333 

287682O72451780 

A- 

A- 

-B-l-C-l  D-l-E-l-F-1  G 

-b4-c-di-e-f-i-g 

=A-H'+^-K-l->H.'- 

( I? ) 

monftration  doth  prove  the  Space  CAVENto  be  equal  to  his 
fuppofed  quantity  *,  and  then  refers  you  to  Caveleriuh  method 
of  Indivifibles,  a  Book  I  have  not  yet  feen.  Which  briefly  I 
conceive  may  be  thuseafily  demonftrated. 

It  is  manifeft  that  all  the  Perpendiculars  let  fall  from  the 
Ordinates  (  Handing  upon  the  prolonged  fide  DB)  to  the  Bafe 
CD,  doth  not  only  defcribe  the  Curve,  but  would  alfo  fill  the 
whole  Hyperbolical  Space,  were  Number,  and  the  Curve  Defir 
nitive.  And  thofe  Perpendiculars  let  fall  from  the  Ordinates 
(  ftanding  upon  DB  prolonged,  numbred  20,  30,  40,  50,60, 
&c.  )  doth  divide  the  Bafe  DC,  in  1  ;  £■  I  ',  &c.  And  where- 
as the  Diagonal  CB,  by  croffing  all  thofe  Perpendiculars  doth 
give  the  firft  Term  of  an  infinite  Series  between  the  Root  or 
Side  AC  (=NH)  and  the  length  of  each  of  thofe  Perpendi- 
culars ftanding  upon  the  Bafe  CD.  Therefore  to  know  the 
Hyperbolical  Space  CAVEN,  divide  the  Parallelogram  CH, 
making  it  the  firft  Term  of  an  infinite  Series  by  the  Ratio  of 
NB  —  NC  to  NH  in  infinitum,  and  each  of  thofe  Quotes  or 
Proportional  Numbers  by  1,2,3,4,5,6,7,8,  &c.  alio  in  infini- 
tum. The  Quotes  of  all  the  laft  Divifions  added,  will  give 
you  the  Area  of  the  Hyperbolical  Space  CAVEN,  and  fofor 
any  other  Curvilined  or  Hyperbolical  Space  ftanding  upon  the 
Bafe  CD,  as  by  the  Calculation  following. 

And  before  I  handle  any  of  the  other  two  Curvilined  Spacer 
differing  in  form,  and  yet  equal  to  this  Hyperbolical  Space 
CAVEN,  we  will  exemplify  this  Demonftration  in  Operation,, 
and  the  figurative  work  thereof,  (hall  be  the  work  of  the  next 
Chapter. 


C  H  A  P.     V. 

LET  it  be  required  to  calculate  the  Area  of  the  CurvilinecT 
Trapezia  or  Hyperbolical  Space  CAVEN  to  15  Places, 
and  hereafter  you  will  fee  it  done  to  25  Places  according  to 
Dairy's  Rule  in  Chapter  VII.  but  with  greater  dispatch. 


The:   } 


f 


(  14) 

Thus  by  the  Calculation  do  you  find  the  Area  of  the  Curvi- 
lined  Trapezia  or  Hyperbolical  Space  CAVEN  to  the  1 5  place 
to  be  5p3 147 1805 59945  ecIual  t0  tt}e  Curvilined  Trapezia 
ZEVAB,  and  alfo  equal  to  the  Curvilined  Trapezia  or 
Hyperbolical  Space  ns-AVX,  the  Calculation  of  the  Areas  of 
any  part  of  thefe  two  latter  (hail  be  (hewn  hereafter  which  will 
differ  in  Operation,  yet  bring  out  the  fame  Number,  and  in. 
Calculating  the  laft,  we  (hall  ufe  Dairy's  Diredions. 

It  having  been  before  fhewn  that  the  Hyperbolical  Space 
2EVAB  equal  to  the  Curvilined  Trapezia  or  Hyperbolical  Space 
CAVEN  is  equal  to  the  Curvilined  Trapezia  or  Hyperbolical 
Space  YGFEZ,  that  therefore  the  faid  ZEVAB  is  a  Space  or 
Quantity  to  reprefent  the  Logarithm  of  2.  So  thenthe  afore- 
faid  Number  693 147180559945  is  an  Hyperbolical  Logarithm 
of  2.  And  having  the  Logarithm  of  2.  you  have  alfo  the  Lo- 
garithm of  all  the  Powers  of  2. 

^  And  by  this  Calculation  you  have  not  only  gotten  the  Loga- 
rithm of  2,  but  gained  alfo  the  Logarithm  of  3,  for  if  you  add 
all  the  Quotes  marked  with  this  Afterifm  (*)  the  Addition  of 
them  (hall  be  the  half  Summ  of  the  Hyperbolical  Logarithm  of 
3-  agreeable  to  the  4th.  Confeftary  of  the  4th.  Propofition,and 
firft  inference  on  the  5  th.  Propofition  of  faid  James  Gregory* '$ 
Exercitationes  Geometric*,  from  whence  Ms  plain  that  Michael 
Dairy  had  his  Rule,  as  will  appear  more  manifeft  after  we  have 
contemplated  the  two  other  Curvilined  Trapezia:  or  Hyperbo- 
cal  Spaces  ZEVAB  and  nsAVX. 

The  Addition  of  the  Quotes  marked  with  *  make 
549306144334055  which  doubled  is  1098512288668110  for 
the  Logarithm  of  3 , and  now  having  gotten  the  Logarithm  of  3, 
you  have  alfo  the  Logarithm  of  all  the  Powers  of  3,  and  of  all 
the  Compofites  of  2  and  3. 

Again  if  you  (hall  from  50  fubflraft  125,  and  to  that  add 
4.166666666666,  and  from  that  fubftrad  15625  and  fo  on 
throughout,  you  fhall  have  the  Logarithm  of  the  difference  be- 
tween 2  and  3,  or  the  Logarithm  of  1  and  4  or  1  and  ,'corre- 
fpondent  to  the  inference  on  the  $  Prop,  of  Jmes  Gregory^  all 
which  fhall  be  fully  exemplified  hereafter. 

The  Calculation  of  the  Logarithm  of  2,  according  to  the 
Method  before  going  is  the  ground  work  of  all  the  Calculations 
followiogvsnd  I  fhall  only  give  the  Calculation  of  one  more 

Space 


(  i5  ) 

Space  to  reprefent  the  Logarithm  of  3  after  that  Method,thougl* 
we  have,  you  fee,  gotten  already  the  Logarithm  thereof,  but 
fuppofing  we  had  not,  and  were  to  find  that  firft  according  to 
the  faid  Method, 

From  F  you  let  fall  a  Perpendicular  as  FO.  So  is  the  Curvi- 
lined  Trapezia  or  Hyperbolical  Space  AEFOC  equal  to  the  Cur- 
vilined  Trapezia  or  Hyperbolical  Space  AEF^B  for  the  Oblong 
AEOC  as  before  fhewn  will  be  equal  to  the  Parallelogram 
£F^B,  and  the  Gurvi4ined  Triangle  AEWL  common,  to  both 
Parallelograms.  Therefore  the  Curvilined  Trapezia  AEFOC 
equal  to  the  Curvilined  Trapezia  or  Hyperbolical  Space  AEF$B, 
to  calculate  whofe  Area  which  will  be  the  Logarithm  of  3  you 
proceed  as  followeth,  which  Work  is  but  partly  done,  to  (hew 
the  way  thereof,  the  Logarithm  of  three  being  hereafter  done 
to  25  places ;  but  with  far  greater  difpatch  than  this  Method 
will  permit. 


I  6666(56666666666 

II  222222222222223 

III  . 58755432098765 

IV  .49382716049382 

v   .26337448559671 

VI  . 1463 191 5866484 

VII  ..8361094780848 

VIII  ..4877305288829 

IX  ..  2890254985973 

X  ..  I7341 5*991 5$3 

XI  ... 1051001813081 

XII  ...642278885771 

I  have  but  gone  twelve  ftepsin  the  Calculation  of  the  Lo- 
garithm of  3  after  faid  Method,  which  will  if  it  were  added. 
but  give  the  Logarithm  of  3  to  five  places.  I  have  left  it  un- 
finifhed  for  the  Exercife  of  thofe  who  fliall  take  delight  hej^inr 
and  finifh  it  throughout  to  the  intent  of  making  the  Logarit^n 
of  3  .to  fifteen  places  accord  ing  to  this  Method.  By  adding  the 
Quotes  of  fo  much  as  is  done  the  firft  five  figures  will  be  10986 
correfpondent  to  the  Logarithm  of  3}  this  Method  being  feme- 
what  flow,  I-  (hall  not  Calculate  the  Logarithms  of  any  other 
Numbers  according  to  it.  And  by  thefe  two  Examples  the 
Reader  may  fee  enough  to  calculate  any  other  Curvilined  %a- 


666666666666666 
444444444444444 
296296296296296 
197530864T97530 
131687242798352 
87791495 198902 

585276634^5934 
39018443310630 

26012294873752 

^34*5:^915834 

1 1 56 1 019943 888 

7707346629258 


(  pO 

pezia  or  Hyperbolical  Space  {landing  onely  in  or  upon  the  Bafe 
CD  equal  to  any  Hyperbolical  Space :  reckoned  in  the  prolon- 
ged fide  DB. 

Andfo  we  will  contemplate  in  the  next  Chapter  the  Affecti- 
ons and  Properties  of  the  Hyperbolical  Space  or  Curvilined 
Trapezia  AseYC  equal  to  the  Curvilined  Trapezia  or  Hyperbo- 
lical Space  ABZEV. 


CHAP.    VI. 

BEforc  we  (hew  how  to  calculate  any  part  of  the  Curvilined 
Trapezia  or  Hyperbolical  Space  As®tc,  equal  to  the  Cur- 
vilined Trapezia  or  Hyperbolical  Space  AVEZB.  We  will  in- 
fert  Tables  to  illuftrate  the  1,2  and  3  Propofitions  of  James 
Gregory**  ExercitatioHes  Geometric*. 


7he 


1 


N 


N 

II 


(  *7  ) 

j- p- ~ j2 q^wonw>  w 

h    M  Mli 

41  \o  \o  ^O    COO\  N^l 
CO<l  ua   O    -<   N*  ^-^ 

CO  0\  ty  v^l 

The  Rule  to  find  Z. 

The  Rule  to  find  Z  is^     S    k% 

5-325=325:5 ::  5:  -r        ^ 

66666666  Impares  ==  ^  ^> 

"s   «  aT  ^   a   w  3? 

The  Proportion  to  find  Z     ►_,  "-^ 

25-525=1875: 25 :  :24:—  ^ 

33333333  Pares  =  Z^^  ^ 

"Th'e^eir^csgggi      |: 

25-625=1875:25::*    ffi    u     < 

25:  =  33333333  ^ 
Equalis  Paribus 

<---- -nr -T~r7"r^  ^ 

-T-l-ri-rl-rl^l^i"1"  vST 

The  Proportion  is  <^     ^ 

5-1-25=75:  5::  5:  =  33333333    ~    ST. 

Equalis  Exceffui  Imparium  fupraom- 
nes  Pares. 
l— "    "v  ■ r"r  "1 


« 


TheProportion  is  ^  e^ 

<-i2S=37^:  5  :  •  S  :  -—66665666 

Equalis  Paribus 


--\r 


"+'1+1  +  1  +  1+1  + 


2A+  2G+  *E+  *H-.  2l-h  ^    ^ 


The  Explanation  of  the  foregoing  Table. 


This  Table  confifteth  of  Eight  Columns,  The  firft  is  a  fup- 
pofed  literal  Rank  of  Quantities  continually  proportional.  The 
fecond  is  of  Numbers  correfpondent  to  the  firft  in  a  Ration  as  2 
is  to  1.  Or  5  to  2,5.  What  2,5  ?  And  fo  fuppofed  continued  in 
infinitum,  with  the  Rule  how  to  find  out  the  whole  Summ  of 
thofe  Numbers  fo  continually  proportional. 

The  third  Column  flieweth  how  to  find  the  whole  Summ  oi 
the  odd  Quantities  or  Numbers. 

The  fourth  teacheth  how  to  know  the  whole  Summ  of  the 
even  Quantities  or  Numbers. 

The  fifth  telleth  how  to  find  the  whole  Summ  of  the  Diffe- 
rence of  the  Quantities  of  the  firft  Column. 

Thefixth  fuppofeth  A-B+C--D-HE,  and  teacheth  how  to 
find  the  Solution  thereof. 

The  feventh  fuppofeth  A  |  -B-C~|  D-E,  and  givethaRuIe 
to  refolve  the  fame. 

Theeigth  and  lafl  fuppofeth  the  whole  Rankr  firft,  Affirma- 
tive, and  the  fecond  evenly  or  alternately  lefs :  And  giveth  a 
Solution  thereof 

A  further  Explanation  of  this  Table  will  be  when  we  come  to 
calculate  the  Area  of  any  part  of  the  Curvilined  Trapezia 7 
AS0YC* 

The  Second  Table. 


0  I      -Ui 

*T3  PC     <L> 

<L>  *-*    C3 

.5  2  c 

^  £   u.   e 

ID  ft           O 

o  .2  **=  '"* 
U 


3  n5"vfiS 


I 

10 
20 

30 
40 

So 
60 
70 
80 
90 
10 


II 

10 

s 

333333333 

2,5 

-2 


11 

QJ     O 

S-<     P-( 


.,tf<J5<56<J55  o  g 
1,42857143  §  I 

1,25  «    W3 

1,11111111    |'| 
I.  J 


III 

OO 

5 

666666666 

7»5 
8, 

8,33333333 

8,57142857 

8,75 
8,88888888 


IV 


Sf  J* 


o 


r    ^  12 

--;>     -U»      C* 


t  This  Table  confifteth  of  four  Columns,  The  firft  is  equal 
spaces  numbred  in  the  fide  DB  prolonged,  or  the  Tangents 
greater  than  Radius.  The 


(  19) 

The  fecond  flieweth  the  length  of  the  Perpendiculars  land- 
ing upon  the  fldeDB  prolonged;  which  are  Tangents  left 
than  Radius,  and  by  the  Tops  pafs  the  Curve  or  Hyperboli- 
cal line. 
The  third  Column  is  the  Arithmetical  Complements. 
The  fourth  Column  flieweth  what  Proportion  the  fecond 
Column  hath  to  Radius. 

The  Re&angleor  Parallelogram  of  the  firft  and  fecond  Co- 
lumn is  equal  always  to  the  fquare  Af). 

This  Table  is  of  ufe  to  find  Points  to  defcribe  the  Curve  or 
Hyperbolical  line,  or  to  examine  if  the  Curve  pafs  through 
fuch  points  as  the  Table  mentions. 

The  makifigofthis  Table  hath  been  formerly  (hewn,  when 
:  it  was  taught  how  to  defcribe  the  Curve. 

We  now  come  to  fhew  how  to  make  a  Table  to  find  the 
length  of  the  Bafes  of  the  Compound  Curvilined  Trapezias,  or 
Hyperbolical  Spaces. 

We  call  that  a  Compound  Curvilined  Trapezia  or  Hyper- 
bolical Space,  when  AC  is  in  the  Middle  of  that  Bafe. 

So  AC  ftanding  upon  the  Middle  of  T1X  hath  Perpendiculars 
or  Sides  H%  and  XV,  fois  the  Curvilined  Trapezia  il^AVX^to 
be  hereafter  underftood  a  Compound  Curvilined  Trapezia  or 
Hyperbolical  Space,  and  will  be  Ihewn  as  followeth  to  be  equal 
to  the  aforefaid  Spaces  CAVEN,  and  to  AVE2B  for  the  Loga- 
rithm Space  of  2. 

And  the  Compound  Curvilined  Trapezia  A0AVEN  will  be 
equal  to  the  Curvilined  Trapezia  AVFOC,  and  to  AVF#R  for 
the  Logarithm  Space  of  3 . 

The  Compound  Curvilined  Trapezia  or  Hyperbolical  Space 
nsAVX  we  may  prove  to  be  equal  to  AVEZB  thus,  by  the  43  of 
the  firft  of  Euclid  the  Parallelogram  CK  is  equal  to  K  5,  and 
the  Curvilined  Triangle  AVK  common  to  both,  fo  then  is 
AVKXC  equal  to  AV5B.    And  the  Parallelogram  n  t  eq^l  to 
the  Square  VZ,  and  the  Curvilined  Triangle  s  f  A  equal  to  the 
Curvilined  Triangle  V^E,  and  fo  the  Compound  Curvilined 
Trapezia  lis  AVX  equal  to  the  Curvilined  Trapezia  AVEZB 
for  the  Logarithm  Space  of  2.  For  by  the  4th  Table  following, 
look  what  Proportion  the  Perpendiculars  or  Sides  of  the  Com- 
pound Curvilined  Trapezias  have  one  to  the  other,  the  like  | 
Proportion  have  the  Sides  or  Perpendiculars  of  the  other  two  * 
Curvilined  Trapezias.  D  2  So 


(    20    ) 

So  in  this  Compound  Curvilined  Trapezia  Hs  and  XV  the 
Sides  or  Perpendiculars  are  in  t  Proportion  as  2  is  to  1  defcen- 
ding,  or  as  1  is  to  2  afcending  \  fo  Hkewife  in  the  Curvilined 
Trapezia  CAVEN  (equal  to  the  aforefaid  Compound  Curvi- 
lined Trapezia  TlsAVX)  the  Side  or  Perpendicular  NE  is  double 
to  CA.  And  alfo  in  the  Curvilined  Trapezia  AVEZB  (equal 
as  [before  to  the  Compound  Curvilined  Trapezia  l)  ~AVX)  the 
Side  or  Perpendicular  B  A  twice  ZE  as  before  taught. 

Thus  by  the  Ratio  of  the  2  Tables  following  may  you  make 
a  Compound  Curvilined  Trapezia  equal  to  either  of  the  other 
two  Curvilined  Trapezias  or  Hyperbolical  Spaces,  and  the 
calculating  the  Area  of  the  Compound  Curvilined  Trapezias 
will  be  found  to  be  of  far  greater  Difpatch  than  the  former 
Method,  by  which  we  (hall  makeufe  of  Dairy*?,  Rule,  or  rather 
the  learned  James  Gregorf%  from  his  firft  Inference  on  his  5 
Propofition. 

We  come  now  to  infer t  the  third  Table,  which  is  a  Table  of 
Ratios  to  find  the  Length  of  the  Bafes  of  the  Compound  Cur- 
vilined Trapezias. 

:  You  may  note,  that  in  all  the  three  different  forts  of  Curvi- 
lined Trapezias  or  Hyperbolical  Spaces  equal  the  one  to  the 
other>  if  on  the  Middle  of  their  Bafes,  you  fhall  ered  Perpen- 
diculars to  touch  the  Curve  the  greater  part  or  fegments  in  each 
is  equal  to  either  greater  fegment  of  the  other,  and  fo  is  the 
leller  part  or  fegment  of  the  one  equal  to  the  lefler  fegment  of 
either  of  the  other. 


t 


Th 


(    21    ) 


The  Third  Table. 

Being  a  Table  of  Ratios  to  find  the  Length  of  the  Bafes  of  th? 
Compound  Curvilined  Trapezias  or  Hyperbolical  Spaces. 


I 

5 


m 

fZD 

I  20 

CZD 

1*5 

'   13° 
<35 


II 
DC 

10 
2D 

20 

DC 

10 
$D 

30 

10 
40 

ic   : 
5o 
10 
60 


III 

:  DC 

:  10 
:DC 

;  :   10 

:  DC 

; :.  10 
:  DC 
:   10 

:  10 

:  10  : 

:  10  : 

:  10  : 

:  10  : 

:  10  : 


IV 
DX 

, 666666666 ' 
DH 
h  3333: 

5 
D\ 

*5 

4 
16 

33333333?! 
1  666656666 f 

285714285  r 

1.714285715  L 


r        vi 

r  C  Length  of 
<     the'Eafe 

1^666666666 

10 


<fi2 

{'333333333 

!  I42857I430 


This  Table  confifteth  of  fix  Columns,  The  firfl  four  fhe  w  the 
Proportion  or  Ratio  to  find  the  Lengths  of  the  Bafes,  and  the 
Number  in  the  fixth  Column  is  the  Length  of  the  Bafe  for  fo 
many  Spaces  as  the  fifth  Column  fignifies. 

And  by  the  fame  Reafon  you  may  find  the  Lengths  of  the 
Bafes  for  any  other  Curvilined  Trapezia  or  Hyperbolical 
Space. 

Thus  is  666666666  of  the  fixth  Colurpn  (the  difference  of  the 
two  firft  Numbers  in  the  fourth  Column)  the  Length  of  the 
Bafe,  for  the  Curvilined  compound  Trapezia  or  Hyperbolical 
Space  to  reprefent  the  Logarithm  of  2. 

And  10  the  Length  of  the  Bafe  for  3r  fo  is  12  for  4 :  and 
xi  333333333  for  5,  and  fo  is  1428571430  for  the  Length  of 
the  Bafe  for  the  Hyperbolical  Space  for  6.  And  thus  may  you 
do  for  any  other  Space  or  Number.  *S 

The  Numbers  in  the  fourth  Column  for  2,  3, 45  5,  5,  &c. 
are  in  Proportion  as  ^,V5  TVs  7  vj  &c*  And  added  are  equal  to 
twice  Radius  or  20  —  D  5 . 

We  proceed  next  to  (hew  how  to  make  a  Table  of  Ratios  to, 
find  the  Lengths  of  both  the  Perpendiculars  (or  fides)  of  the 
Compound  Curvilined  Trapezias; 

Tk-. 


(    22    ) 

The  "Fourth  Table. 

Being  a  Table  of  Ratios  to  find  the  Length  of  both  the  Per- 
pendiculars or  Sides  of  the  Compoun  Curvilinedd  Trapezias  or 
Hyperbolical  Spaces' 


1 

Ccd—    ex 

\\o  —333333333  ■ 
}CD+CU(=CX)  : 
C  io -{-333333333    : 

CCD— CN 
ho—  $ 

C  lo  -J-  5 
•CD— C® 
io  —  6 

>CD+C»(=DsO 
•IO  -|-    6 

cd — ca 

io  —  666666666    ■ 

iCD-r-cnp(=:ca; : 

10  -1-    666666666 


II 

s  XD  :DB: 
:  666666666  : io  : 
:  Dfl  :DB: 

:i333333333:io  : 


=ND 

=  % 

:DII 
=  15 

=  4 

=D« 

-\6 

=aD 


:DB: 
:  io  : 
:DB: 

:  io  : 

:DB: 
:  io  : 
:DB: 
:  10  : 
:DB: 


=  333333333:io  : 
:D"?  :DB: 

: 1 6666666 66: 10  : 


III     IV  V 

:DB:Ds=XV 
110:15        ) 
:DB:I2     J  2 
:io  :  7,5 

:DB:D2=:NE 

:  10  : 20       ? 
:DB:  ^0    f  3 
:io  :    666666666 

:DB:DR=sII 
:  10  .-25      "7 
:DB:  r«»  >■  4 
:io  :6z$    ^ 

:  DB :  D*=rFa 
:  10  .-30      ~) 
:DB:>";~     > 
: 10  :   6      3 


% 


ft 


This  Table  confiftcth  of  five  Columns,  The 

firft  contains  the  Quantities  and  Numbers  of 

the   firft  Term  in  the  Proportion  ;  The  fe- 

cond  Column  the  Quantities  and  Numbers  of 

the  fecond  Term  in  the  proportion ;  The 

third  Column  the  Quantities  and  Numbers  of 

the  third  Term  in  the  Proportion  $    The 

fourth  Column  the  Quantities  and  Numbers 

of  the  fourth  Proportional  Number  or  Term, 

J>*w  wherein  are  Numbers  for  the  Length  of  both 

the  Perpendiculars  for  2, 3 , 4, 5,  &c    The 

fifth  Column  is  the  Numerical  Order  of  the  Compound  Curvi- 

lined  Trapezias  or  Hyperbolical  Spaces  of  2, 3?  4*  h  &c* 


35 

583333333 
40       ? 

5714285715  1 

5^5     S 
So       ? 

Sf5555555(9 
55       Y 


10. 


And 


And  by  the  fame  Ratio  you  may  find  the  Lengths  of  both  the 
Perpendiculars  for  any  other  Compound  Curvilined  Traperias 
to  reprefent  the  Logarithm  of  any  other  Number. 

By  thefourth  Column  you  may  perceive  the  Perpendiculars 
or  Sides  (  of  the  Compound  Curvilined  Trapezias  or  Hyper- 
bolical Spaces )  are  in  fuch  Proportion  the  one  to  the  other, 
as  the  Number  they  reprefent  are  to  Unity. 

That  is  to  fay.  In  thefCompound  Curvilined  Trapezia- 
n^AVX  to  reprefent  the  Logarithm  Space  of  2.  the  Perpendi 
ular  XV  is  to  fls  as  2  is  to  1. 

And  in  the  Compound  Curvilined  Trapezia  A© AVEN  to  re- 
prefent the  Logarithm  Space  of  3,  the  Perpendicular  NE  is  in 
proportion  to  A©  as  3  is  to  1. 

And  fo  the  Perpendiculars  of  the  fourth  Logarithm  Space  as 
4  to  1.  And  of  the  fifth  Space  as  5  to  1,  &c.  as  by  the  fourth 
Column  of  this  fourth  Table  appeareth. 

And  the  Perpendiculars  of  both  the  other  forts  of  the  Curvi- 
lined Trapezia  or  Hyperbolical  Spaces  are  likewife  in  the  Very 
fame  Proportion  the  one  to  the  other,  as  you  may  note  from 
what  hath  been  faid  before  of  them. 

By  thefe  Tables  and  by  what  hath  been  faid  formerly ;  thefe 
three  Curvilined  Trapezias  have  the  fame  Properties  and  Affec- 
tions as  thofe  have  in  an  Hyperbola  derived  from  the  Section  of 
a  Right  Angled  Cone. 

We  (hall  now  therefore  come  to  calculate  fome  part  of  this 
latter  HppcrbolicaJ  Space  before  we  fhew,  how  to  do  it  all  at 
once  j  that  isoftheHperbolical  Space  AOAVEN  to  calculate 
the  Area  of  the  Space  A®  AC  which  isequalas  before  fliewnto 
the  Space  of  5 VAB.  And  when  we  have  (hewn  to  calculate  this 
part,  we  fhall  from  this,  and  what  hath  been  taught  how  to  do 
the  other  Part  come  to  derive  Dairy's  Rule,  or  rather  James  Gre- 
gory's.,  which  is  comprifed  in  thefirft  Inference  on  his  %th  Pro- 
portion, m^ 

CHAP.     VII. 

\KT  E  have  in  the  $tb.  Chapter  calculated  the  Area  of  the 
V  V    Curvilined  Trapezia  or  Hyperbolical  Space  CAVEN 
equal  to  AVEZB  for  the  Logarithm  of  2. 

In 


(  H) 

In  this  Curvilined  Trapezia  CAVEN  all  the  Perpendiculars 
ftanding  upon  the  Bafe  CN  are  each  more  than  Radius  CA  (  or 
greater  than  the  Tangent  of  45^.  oo\)  being  ft  ill  afcendingand 
affirmative,  and  thefore  by  the  1ft.  Table  to  be  continually  ad- 
as  by  the  Calculation  thereof  is  alfo  manifeft. 

We  new  come  to  calculate  the  Curvilined  Trapezia  C  A© A 
part  of  the  Curvilined  Trapezia  Cr0® A  equal  to  AVEZB. 

In  this  Curvilined  Trapezia  C  A©A  (equal  to  B5VAJ  all  the 
Perpendiculars  ftanding  upon  the  Bafe  CA  are  each  leflerthan 
the  Radius  C  A  being  {till  defcendiug  and  negative,  and  there- 
fore to  be  handled  by  the  firfb  Table  accordingly. 

The  Bafe  C  A  is  equal  to  CN  of  that  Space  calculated  as  be- 
fore in  the  $th-  Chapter. 

If  by  the  Vertex  A  you  draw  a  Parallel  te  the  Diagonal  CB 
as  ZAT  it  is  a  Tangent  to  the  Curve  touching  it  in  the  point 
A,  and  AB  doth  cut  all  the  Perpendiculars  contrary  wife  toCB. 
For  CX=Xt$  is  not  equal  torn=Tl^  but  II 6  is  equal  to 
K^=KBbecaufeCn  is  equal  to  CX,and  the  Angle  TlT6  equal 
to  the  Angle  KB  nf  fo  therefore  by  the  1  and  4  Table  all  the  Per- 
pendiculars ftanding  upon  Cr  are  leffer  than  Radius.  And  fee- 
ing by  the  fixth  Column  of  the  firft  Tableland  alfo  by  the  4^. 
of  the  fourth  Table  we  may  find  the  Length  of  A0,  Therefore 
to  know  the  Area  of  C  A©A  making  C©  the  firft  Term  of  an 
Infinite  Series  in  a  continual  Proportion?  as  CA  is  to  CA  that 
is  as  50  to  25  what  25  ?  facit  125,  as  in  the  Infinite  Series  of 
Numbers  continually  Proportional  for  the  Calculation  of  the 
Logarithms  of  2  in  Chap,  5  you  do  therefore  as  there  faidfrom 
50  ( of  the  fecond  part  under  the  Title,  the  Quotes  to  be  ad- 
ded )fubftrad  125,  and  to  that  add  4.16666666666666,  and 
from  that  fabft raft  15525  and  fo  on  throughout,  you  fhall  have 
405465 ip8 108165  for  the  Area  of  the  Space  CA0A  equal  to 
BAV5.|And  thus  may  you  find  any  other  part  of  CTHA. 

^Ve  (hall  fhew  how  to  do  it  for  CII2A  and  GXVA,becaufe  from 
t&em  we  (hall  derive  Dairy  s  Rule  or  rather  James  Gregory's,  for 
from  them  we  have  derived  and  calculated  the  Logarithm  for 
2  to  twenty  five  places,  as  by  the  Calculation  following  next 
after  this  wiU  appear. 

Now 


Now  to  Calculate  the  Area  of  the  Curvilined  Trapezia  or  Hy5 
perbolical  Spaces  CtirA  and  CXVA,you  make  Cor  CH  the  Firfl 
Term  of  any  Infinite  Series,  and  the  Second  Tei-m  in  fuch  a 
Ratio  as  FIG  is  to  CA  for  your  Proportionals  of  your  Infinite 
Series,  and  fo  proceed  on  as  in  Chap.  V.  and  as  here  appeareth. 


The  Infinite  Series  of  TV 'um- 
bers Proportional. 

a  333333333333333-   I     A  - 
a*.  1 1 1 1 1 1 1 1 1 1 1 1 1 1 1  •    II    B  - 

aaa  .  37°37°37°37°37-  M  C  > 
a  4  .  12345*78012345  .  IV  D 
a  r  . • 41 1 5226337448  .  V  E 
a  6  . . 1371742112483  .  VI  F 
a  7  ...457247370827.  VII  G 
A*  ...  152315790275  .VIII  H 
a3  ....50805263425.  IX  I 
al°    ....  16935087808.    X    K 

a11    5645029269.   XI  L 

a12    1881676423.  XII  M 


The  Quotes  to  be  added 
for  CXVA. 

333333333333333  +  A 
-   55555555555555 +B 

12345*79012345  -i-c 
.  3086419753086+  D 
.823  045  267489  -|-E 
,.  228623685414+  F 
.  65321052927 -|-G 
•  190S1973784  +  H 
..  .  5645029269+  I 
..1693  508781 +  K 
....513184479+L 
....  156806369  4- M 


We  have  but  gone  through  Twelve  Steps  of  this  Calculati- 
on, to  (hew  the  manner  thereof  -,  but  fhould  you  proceed  to  go 
through  it  till  it  works  off,  as  in  Chap.  V.  you  may  have  both 
the  Segment  CXVA,  and  CTlrA  ;  for  if  you  finilh  the  Calcula- 
tion, and  add  up  all  the  Quotes,  that  Sum  will  be  the  Area  of 
CXVA,  and  be  found  405465 108 108165,  asinChap.  V.  and  is 
equal  to  the  greater  Segment  WNHE  2  in  the  Curvilined 
Trapezia  or  Hyperbolical  Space  CNHE  2  A,  and  alfo  WNHE  J 
is  equal  to  C  AG  A  equal  toAB5  V.  <*\ 

And  if  from  333333333333333  you  fhall  fubflrad  the  Se- 
cond Number  555555555555555^  and  to  that  add  the  Third 
Number  2345679012345,  and  then  from  that  fubflrad  the 
Fourth,  and  fo  add  and  fubftradt  according  to  the  Signs  -]-' 
and  —  throughout,  you  will  have  .  .  287682072451780  for 
the  Area  of  the  lefler  Segment  of  the  Compound  Curvilined 
Trapezia  wn  TAJ  that  is  the  Area  of  CFI2A  equal  to  CWDA 
equal  to  V  5  ZE.  E  And 


V 


(  So  ) 

And  you  have  not  only  gotten  by  this  Calculation  the  Area 
of  each  Segment  feparately,  and  fo  ccnfequently  the  Area  of 
the  whole  Space,  by  Addition  of  thofe  Two,  but  you  have  alfo 
obtained  the  half  of  the  whole  Area  at  once,for  if  you  fhall  (cor- 
refpondent  to  the  Column  of  the  Firft  Table  )  add  the  Num- 
bers with -|- affirmative,  they  will  give  you  half  the  Area  of 
the  compound  Curvilined  Trapezia  XnrAV  for  the  Logarithm 
of  2,  which  you  will  fee  prefently  exemplified  and  done  to  25 
places  :  And  this  is  the  Summ  of  James  Gregories  Inference  on 
his  Fifth  propofition  of  his  Exercitationes  Geometric*  ;    and  fo 
agreeable  to  the  Rule  delivered  to  me  as  before  declared  by 
Mich.  Dairy  :  Having  acquainted  feveral  Perfons  with  Dairy's 
Rule  in  page  1,  and   (hewn  to  them  fome  figurative   work 
thereupon  in  Order  to  make  a  Logarithm,  I  was  notwithfland- 
ingfome  time  through  inadvertency  almoft  difcouraged  of  ever 
knowing  how  to  Cube  in  infinitum  fach  a  Number  as  there  fpo- 
ken  of,  neither  did  any  of  thofe  to  whom  I  had  communicated 
the  fame,  take  any  fuch  notice  thereof  (that  1  knowj  fo  as  to 
do  it.    And  now  1  come  to  fhew  how  I  overcame  that  difficulty 
of  Cubing  a  Range  of  Figures  for  25  Places,  which  he  told  me 
I  mull  do  in  Infinitum^  before  I  could  make  the  Logarithm  of  fo 
many  places  j  and  to  remove  this  flumbling  Block  (  I  do  con- 
fefs)  took  up  fome  time  ;  for  Dairy  had  not  then  told  me  a 
word  of  fuch  an  Authour  as  James  Gregory  ^  and  I  had  not  known 
his  Works,  but  for  John  Collins,    fome  years  after  Dairy 
Death  •,  but  before  I  ever  met  with  Gregory* s  Book,  I  had  oi 
tained  my  defire  to  Cube  in  Infinitum  Twenty  Five  Figure*, 
That  is  Twenty  five  3  by  dividing  by  9  continually,  as  in  ** 
Calculation  following,  to  find  the  Logarithm  of  2  allatc_w, 
which  manner  difpatcheth  the  Calculation  much  more  fpeedy 
than  the  Method  of  Calculation  in  the  Fifth  Chap. 

And  now  the  Reafon  of  Cubing  Twenty  Five  3,  by  dividing 
onl£oy  9 doth  follow. 

For  as  muchas-D^V/s  Rule  before  declared,  to  make  the 
Logarithm  of  2,  doth  bid  you  to  2,  add  i,  and  from  2  fub- 
ftratt  1  •,  fo  fhall  there  be  a  Refult  orFra&ionof  |,:  and  then 
divide  1  or  100,  000, 000,000, 0000  by  J,  whofe  Quotient  is 
333333333333333*  which  Cube  in  Infinitum^  it  had  been  as 
/  much  as  if  he  had  faid  Cube  |  fractionally,  which  is  ~T, 
and  divide  ioooooooqoqqoooqo  by  |7  the  Quotient  will 

be 


(  *  > 

be  37037037037037  for  the  Cube  of  a  or  &  as  in  the  Opera- 
tion before  going-  Now  for  as  much  as  you  would  Cube  the 
Number  for -j-,  viz.  333333333333333  (which  is  i,onoo, 
000,  ooo,  000,0000,  divided  by  3  )  it  is,  as  if  you  fhouki  fay 
as  27  to  1 000000000000000:  what  1  ?  the  Quotient  will  be 
37037037037037  for  the  Cube  of  a  or  333333333333333, 
as  before.  Now  if  you  fhall,  as  in  the  Operation  before-going, 
fet  down  333333333333333  (which  is  equal  to  f,)  you  have 
no  more  to  do  but  to  divide  by  9,  for  that  *  of  '  is  equal  to  \ •?, 
and  therefore  dividing  333333333333333  by  9,  the  Quo- 
tient will  be  370370370370370,  as  before  for  the  Cube 
of  l  or *,  and  feeing  4  X-?  is  equal  to  £,  you  have  no  more  to 
do  but  to  divide  continually  by  9,  and  they  fhall  all  be  Propor- 
tional Numbers  by  7th  of  the  8th  of  Euclid^  and  confequently 
correfpondent  to  the  odd  Powers  •,  for  if  the  Root  be  multi- 
plied by  the  Square,  that  begets  the  Cube,  and  the  Cube  again 
by  the  Square,  that  begets  the  Fifth  Power,  and  fo  on.  So 
here,  for  as  much. as  dividing  by  9  doth  beget  the  Third 
Power  5  if  you  fhall  therefore  continually  divide  by  9>  you 
fhall  have  the  refpe&ive  odd  Powers  accordingly,  as  is  alfo 
manifefl:  by  the  laft  Figurative  Calculation  ;  and  all  is,  for  that 
a  1  doth  neither  multiply  nor  divide,  and  that  |  of  i-  is  equal 
to  2',  and  if  you  fhall  divide  2  4  of  1,  by  9,  the  Quotient  will 
be333333333333333i  which  is  equal  to  4  for  the  Firft  Num- 
ber or  Root,  as  before. 

Now  for  as  much  as  to  make  a  Compound  Curvilined  Tra- 
pezia equal  to  an  uncompounded  :  As  for  inftance,  to  make  the 
Compound  Curvilined  Trapezia  WnrA  j  to  be  equal  to  the 
uncompounded  CAVEN  equal  to  ABZE  —  A0  T  for  the  Lo- 
garithm of  2,  and  to  find  the  Length  of  the  Bafe,  and  both 
the  Perpendiculars,  hath  been  difcourfed,  and  may  be  feen,  as 
in  the  Third  and  Fourth  Table  before-going.  We  come  to 
handle  and  calculate  the  Area  of  this  compounded  Cut%lined 
Trapezia  WTirjA  for  to  make  the  half  Logarithm  of  2 
at  once. 

Seeing  by  the  Sixth  Column  of  the  Third  Table,  the  Bafe 
WI1  h 6666666666666666 ywhofehalfis.  .333333333333333 
for  CW  or  C  n  equal  to  the  Firft  Term  in  the  former  Opera- 
tion (  and  alfo  the  fame  as  Dairy's  Refult  or  Fraction  of  T  k 
and  that  I  muft  divide  in  the  Ratio  of  AC  to  Cfl  or  CW  'in  In- 

E  2  ftnitum,       \ 


(  *o 

finitum^  as  in  the  Fifth  Chapter,  and  alfo  as  in  this  is  fhewn 
and  taught,  for  to  make  the  Infinite  Series  of  Numbers  Pro- 
portional: Itwiil  appear  that  if  I  do  divide  333333333333333 
by  9,  it  will  give  mc  the  Cube  of  the  Firft  Term,  and  fo  divi- 
ding continually  by  9,  will  produce  the  Numbers  appertaining 
to  the  odd  Powers,  as  by  the  large  Calculation  to  25  places 
next  following:  And  feeing  I  am  by  Dairy's  Rule  or  rather 
James -Gregory \  to  divide  each  of  the  Numbers  of  the  Infinite 
Series  bylhe  Indices  of  the  odd  Powers,  it  is  manifeft,  That 
this  Rule  of  Dairy's  is  derivable  from  the  8  Column  of  the  Firft 
Table  ,  for  A-f  B-j-  C  -|  •  D-f  E+  F+  G  +  H  +  I 
And  A  —  B  +  C  —  D-l-'E— F-|-  G  — H-|   I 

doth  make  2  A-~|~  2C~f  2E+  *G-|-  2 1-|-  2L+  2N  : 
And  therefore  every  other  Line  of  the  Quotes  to  be  added  in 
the  former  Operation,  doth  make  half  the  Logarithm  of  2. 

In  making  the  Infinite  Series  in  page  34,  in  Order  to  make 
the  half  Logarithm  of  2,  to  2  5  places  be  very  careful  to  fet  the 
figures  in  their  due  places,  and  to  make  that  Series  you  are  to 
divide  continually  by  9,  which  being  done  throughout,  you  may 
then  prove  your  work  by  Multiplication  in  multiplying  each  line 
by  9,  and  if  thofe  Multiplications  produce  the  foregoing  Num- 
bers you  may  conclude  that  part  of  the  work  to  be  well  prepa- 
red. And  feeing  by  the  dire&ion  over  the  figurative  Work  in 
Page35>  you  are  to  divide  each  of  the  Numbers  in  page  34,  by 
x>  3>  5>  7?  9,  <^x  You  muft  fo  order  the  Quotes  of  page  35, 
that  they  may  lye  in  the  fame  line  or  range  with  their  refpec- 
tive  DividendsorNumbers  in  Page  34 :  for  the  better  Preven- 
ting miftakes,the  letter  Figures  do  reprefent  the  Divifors  pro- 
per to  each  line ;  and  would  you  make  the  Logarithm  of  2,  ac- 
cording to  that  Method  in  page  34  and  3  5,  for  7  or  8,  places 
only,  you  may  very  well  produce  it  in  half  an  hours  time  as  by 
that  Calculation  is  very  perceptible.  And  fome  that  have  had 
thof<L&wo  ftieets  I  formerly  Printed  as  a  Specimen  hereof,  have 
told  mc  they  have  done  the  fame,  and  were  very  folicitous  I 
would  asfoon  as  I  could,  publifti  the  remainder  ,which  at  length 
as  time  and  leifure  hath  permitted  is  done :  and  though  1  have 
not  here  inferted  many  Examples ;  yet  by  what  are  herein  done 
you  may  perceive  how  to  proceed  for  any  other  Number  pro- 
pofed.  And  with  the  diredion  and  reference  in  page  46,  thofe 
/^hat  are  willing  and  curious  herein  may  make  a  Logarithm  for 

any 


( ?J ) 

any  natural  Number  defired.  I  have  not  added  hereto  any  Ta- 
ble of  Logarithms  at  this  time,  and  what  I  may  do  hereafter  in 
order  thereunto  I  do  not  prefume  to  promife.  I  doubt  not  but 
fbme  may  both  examime  fome  Table  or  other,  or  make  by  this 
Method  one  De  Novo,  and  fatisfie  themfelves  about  the  fame, 
and  fome  have  told  me  lince  my  communicating  this  Method 
unto  them,  that  if  the  firft  makers  of  Tables  of  thefe  Numbers 
had  made  them  by  fuch  eafie  ways,  they  did  not  doubt  but  their 
Tables  might  have  been  fomewhat  more  exadt.  Howfoever  it 
pleafed  God  who  is  the  giver  of  every  good  and  perfect  Gift, 
to  raife  and  endue  fuch  men  with  great  ability  and  patience  to 
perform  thofe  Tables  with  fo  much  difficulty  and  labor  as  their 
Methods  did  require,  and  for  common  Ufes  fufficient.  And  with 
fuch  Eagernefs  did  tl»at  Age  embrace  and  purfue  the  Invention 
of  thefe  Numbers  that  VlUch  a  Dutchman  had'exhibitedaTable. 
of  Logarithms  to  io  places  for  iooooo  before  the  Learned  Hen- 
ry Bn£if s  Table,  which  he  had  in  part  done  to  1 5  places,  could 
be  accomplifhed  by  him.  So  exceeding  glad  were  they  of  the  In- 
vention. And  the  Learned  Henry  Briggs  in  his  Epiftle  Dedicato- 
ry to  Our  Mofl  Gracious  King's  Father  when  Prince  of  Wales, 
faith,  that  amongit  the  Antients  there  is  not  found  any  Foot- 
ftepsof  thefe  Numbers;  of  whofe  Conftrudlion  and  Ufes  the 
faid  Henry  Briggs  hath  written  in  his  Arithmetic*  Logarithmica 
molt  learnedly  and  copioufly,  and  now  follows  the  figurative 
part  of  making  the  Logarithm  of  2  1025  Places. 


The 


( i\) 


The  Infinite  Series  or  Numbers  continually  Proportional,  Thefe 
Numbers  are  continually  divided  by  9,  in  order  to  make  the  Hdf 
Logarithm  of  1* 


'Differentia 


Vnitas 
Numeruss 
Propofit.  * 

Summa  . — - 


r 


3333333333333333333333333       1 

370370370370370370370370    HI 
41 1 52263374485 5967078 1 9    V 
•  457247370827617741 1980    VII 
50805263425290860133 1    IX 
56450292694767622370    XI 

6272254743863059152  .XIII 
696917193762563239    XV 

7743  524375i3959  *  5  XVII 
86039 l 5972377324   XIX 

955990663597480  XXI 

106221184844164  '  XXIII 

11802353871574  XXV 

13 11 372652397  XXVII 

145708072489  XXIX 

16189785832  XXXI 

1798865092  XXXIII 

199873899  XXXV 

222082 1 1  XXXVII 

24<*7579  XXXIX 

274J75  XLI 

30464  XLIII 

3385  XLV 

376  XLVII 

42  XLIX 


3 


aaa 
=4X4=1 


r 


fi 


I 


You  may  perceive  that  if  1  be  added  to  2,  and  fubflrafted 
from  it,  itleaveth  aRefultof^,  which  multiplied  into  it  felf, 
maketh-,  and  therefore  thefe  Numbers  are  continually  divi- 
ded by  9. 

Thefe 


<3S) 

Thefe  Numbers  are  Quotes  from  thofe  on  the  oppofite 
Side,  by  dividing  them  by     i,  3,  5,  7,  9,  &c.    and  are* 

JA-h  lC+  'E-h  "G-h  %\-\-  2L+  »N-l--,-c^c.    Correfpon- 


dent  to  the  laft  preceding  Calculation,  which  added  make  half 
the  Area  of  the  Compound  Curvilined  Trapezia  XBTAV  for 
the  Half  Logarithm  of  2  to  2s  Places. 


3333333333333333333333333 

A 

1 2345^79° 1 23456790*  23457 

C 

82304526748971 19341 554 

E 

6532 10529753739630283 

G 

56450292694767622370 

I 

5 1 3 1 84479043 3 4202 1 6 

L 

482481 1341433 13012 

N 

4646 1 1 462  5  083  7  5  49 

P 

45550143383 17407 

R 

452837682756702 

T 

45523364933213 

W 

461 83 12384529 

Y 

472094154863 

48569357496 

5024416293 

52225115^ 

545 1 1063 

5710683 

600222 

63271 

6687 

708 r 

7$ 

8 

"uhSof  T'  {3465735902799726547086160* 

Thus         \ 


' 


(.3«) 

Thus  have  we  Calculated  the  Logarithm  for  2  to  25  Places, 
after  Dairy's  Rule,  or  rather  James  Gregory's,  which  Method 
maketh  far  greater  difpatch  than  that  in  Chap.  V.  for  this 
Calculation  though  to  25  Places,  is  fooner  performed  than 
that  of  1 5  Places  in  Chap.  .V .  as  by  comparing  them  is  very 
perfpicuous  and  manifeft* 

And  now  we  have  exemplified  the  Rule  Dairy  declar'd,  and 
I  am  apt  to  believe  he  had  ftudied  well  Gregory's  faid  Exercita- 
tiones,  though  he  was  not  pleafed  to  tell  any  more  thefreof, 
but  that  others  fhould  take  pains  therein  as  well  as  he,  and 
truly  if  John  Collins  had  not  acquainted  me  with  Gregory's 
Works,  I  had  done  the  Work,  but  not  with  that  fatisfa&ion 
I  met  with  from  James  Gregory's  Books  •,  and  here  you  have  it 
in  a  more  familiar  Difcourfe  and  Dialed  than  that  of  James 
Gregory's,  being  altogether  Analitical,  and  if  any  Letter  or 
Symbol  be  milteken  in  his,  it  is  very  great  Study  and  Labour 
to  find,  and  to  fet  it  to  rights. 

I  find  James  Gregory  hath  calculated  the  Hyperbolical  Loga- 
rithm for  2  in  his  Per  a  Circuit  &  Hyperbola  (Quadrat  ur  a  to  25 
Places,  which  agreeth  with  this  Calculation,  but  to  17  Places 
1  have'not  railed  the  Logarithm  for  2  to  his  Dottrine  in  that 
Book,  but  am  fatisfied  this  Calculation  for  the  Logarithm  of  2 
in  this  Chapter  is  true  to  an  Unite  in  the  25  th  Place,  and  may 
be  in  Two  hours  very  well  Examined  by  any  one  that  will  take 
the  pains  todo  it,  and  they  fhall  find  it  to  be  as  herein  Calcu- 
lated. And  to  this  I  have  the  Concurrence  of  the  moft  inge- 
nuous and  laborious  Mr.  Abraham  Sharp,  who  (from  the  Occa- 
sion of  my  publifhing  formerly  two  fheets  of  the  Praxis  hereof 
as  a  Specimen  )  hath  (hewn  me  his  Calculation  of  the  Loga- 
rithm of  2,  and  fome  others  to  forty  Places,  the  like  I  fuppoie 
not  hitherto  heard  of  or  feen.  Without  all  doubt  Gregory 
found  that  Mercators  Lucky  Invention  of  Squaring  the  Hyper- 
bofa,  was  of  far  more  difpateh  than  that  of  his  Vera  Circuit 
&_  Hyperbola  Quadratura,  or  elfe  he  would  not  have  Writ  up- 
on Mercator  :  But  fo  excellently  hath  Gregory  Illuftrated  Mer- 
cator,  that  a  better  way  of  Squaring  the  Hyperbola  I  fuppofe 
hath  not  «or  oiay  be  found. 


We 


( %1 ) 


°f/'J £ ftKStaiT^  SS  have  theTogaVkhm  of  $,  b* 
afEieS  ;  that  is,  4multiplyed  by  :.  I,  makes  J  :  And 
caufe  that  4X  1 -<\  m™     >  ,  RVithm  of  5,  and  having  got  the  Loga- 

this Method IW k  torn* e  tta  Lo g ^  ^  ^ B  B ^ ^ 

rLth?  °f  thmof  10  S  when  I  have  (hewn  this,  1  (hall  produce  Britfs 
the  ^rt^^nefindcDivifion,  for  that  all  forts  of  Logarithms 

the  Logarithm  for  1  \- 


The  Calculation  of  the  Logarithm  for  1  \ 
to  2*  Places.  ':■■■■.■  .      ,, 

The  Infinite  Series  or  Numbers  continually 
Proportional  5  tbefe  Numbers  are  continu- 
ally divide dby%i. 


miiiiTUUiu****1*1111 
I57i742ii24828S3223594° 
169350878084302867m 
209075 1 58 1 2876897 1 8 
25811747917*31972. 
318663554532493 
39341 1795719*  . 
48569357496  XV 
599621697  avu 
7402737  XIX 
91392  XXI 

1 1 28  xx  in 

*4 


1 
111 

V 

VII 

IX 

XI 

XIII 


Tbefe  Numbers  are  Quotes  from  t\ 
on  the  oppofue  Side,  they  being  di 
dedby  1,  3,  5, 7,9,  &c  ™hich 
ded,  make  half  the  Logarithm 
1  |  to 2%  Places. 

iiriiiiiiiiuiininiiiii 

457247370827617741 1980 

33870175616860573422' 

298678797326812817 

286797 * 99°79244x 
28969414048408 

302624458245 

3237957J6<: 

3527186- 

38961! 

435 

4 


111571775657104877883147 
This  Numb,  is  4  the  Log.  for  1 
22314355*3*4209755766295 
This  Number  is  the  Log.  of  1 


\ 


Differentia        25 


(  38) 

Adding  1  to  1  \ ,  it  maketh  2  \ ,  and  fubflrafted  from  it, 
kaveth  \ ,  which  Refult  maketh  a  Fraction  of  -I  ,  for  2  ' 
being  reduced  into  Fourths,  make  * ,  fo  the  Refult  of  ;  and  % 
(rejecting  the  Denominators )  is  ~  as  above,  which  Squa- 
red maketh  *-!  }  fo  are  thefe  Numbers  therefore  continually 
divided  by  81,  to  make  the  Infinite  Series.  By  Decimals  it 
prefently  Ihewethit  felf  to  be  a  Fra&ion  of 
3 .  Thus  the  Difference  or  Numerator  is  25 
theSummor  Denominator  2,25,  Which 
Decimal  Fra&ion  2^A  is  equal  to  £-« 

To  divide  the  Numbers  on  the  other  fide 
(  to  make  the  Infinite  Series)  by  81,  isea- 
fie  enough,  for  it  is  but  dividing  twice  by 
9,  or  taking  one  Ninth  part  twice,  and  re- 
jecting or  cancelling  the  firft,  _fo  is  it  very 
readily  done,  and  the  whole  Operation  hereof  may  very  well 
be  performed  in  two  hours  time-,  and  thus  have  we  got  the 
Logarithm  for  1  J  to  25  places,  and  now  {hall  proceed  tomake 
the  Logarithm  of  10,  which  is  by  adding  together  the  Loga- 
rithm of  2,  3  times,  and  that  makes  the  Logarithm  of  8,  and 
that  added  to  the  Log.  of  1  \ ,  makes  the  Log.  of  10  ;  and  the 
Logarithm  of  2  fubftradled  from  the  Logarithm  of  10,  leaveth 
the  Logarithm  of  5,  and  is  to  the  fameeffeft  as  is  before. 


Vnitas 
Humerus  1 
Vropofum  ) 

Surnrna 


1,25 


2,2S 


Logarithm 
Logarithm 
Logarithm 
Logarithm 
Logarithm 


of 
of 
of 
of 
of 


2. 


10. 


5* 


tf931471805599453094.1723.21 
2079441 54167983592825 16563 

223  U355  T3H2°97  5  5766295 1 
23025 850929940456840 1799 1 4 
1 60943  79 1 243  4 1 003  746007  5  93 


We  have  now  made  and  exhibited  the  Logarithms  of  2,  5, 
and  io?  and  from  thefe  you  may  make  all  their  Compofites. 

And  nf-w  we  proceed  to  make  the  Logarithm  of  3  to  25 
Places,  'Which  we  (hall  (hew  two  ways,  firft,  all  at  once  from  a 
Compound  Curvilined  Trapezia  or  Hyperbolical  Space  ;  fe- 
condly,  by  a  Competition  of  2  Logarithms,  viz,,  of  2  and  1  \7 
for  that  2  X  i  \  maketh  3,  and  this  latter  we  chiefly  recom- 
mend. Th^  Compound  Curvilined  Trapezia  or  Hyperbolical- 
Space  N  a©  AVE  we  have  in  the  foregoing  Chapter  (hewn  to  be 
ecral  to  the  uncompounded  CAVFO,  andalfo  equal  to  AVF$B, 

we 


( 19  y 

we  fhall  Calculate  the  Logarithm  for  3,  according  to  the  Com- 
pounded Space,  and  by  the  Third  and  Fourth  Tables,  you  may 
know  the  Lengths  of  the  Bafeand  Perpendiculars. 

The  Bafe  N  A  is  10,  therefore  CA  =  5  =  CX.  Now  for  as. 
much  as  Dairfs  Rule  of  adding  to,  and  fubftra&ing  1  from  3^ 
produces  the  half  Length  of  the  Bafe  agreeable  to  the  Third 
Table,  we  fhall  fhew  how  to  Calculate  the  half  of  the  Loga- 
rithm for  3,  as  we  did  for  the  half  Logarithm  of  2. 

Adding  ito3  it  makes  4,  and  fubftradting  1  from  3,  Iea- 
veth  2,  which  maketh  a  Refult  or  Fra&ion  of  \  —  ~. 

Now  dividing  1  and  25  Cyphers  by  4,  the  Quotient  is  5, 
for  the  firft  Term  of  your  Infinite  Series  Now  for  as  much  as 
\  X1,- maketh  \\  the  fecond  Term  muft  be  therefore  ~  of  the 
Firft,  and  fo  on,  as  was  difcours'd  beforafh  making  the  half 
Logarithm  of  2. 

Having  made  the  Infinite  Series  as  followeth,  you  divide  each 
of  thofe  Numbers  (  which  as  before  Taught  are  proportional ) 
by  ii3j$*7)9*&c-  and  thefe  added,  make  half  the  Area  of 
NA$AVE  for  the  half  Logarithm  of  3. 


F  2  The 


(4°) 

The  Infinite  Series  or  Numbers  continually  Proportional.  Thefe 
Numbers  are  continually  divided  by  4,  in  order  to  make  the  \  Log* 
°fi  to  25  Places. 

5  000000000000000000000000 


1 


/ 


ttdtt 

125 

aaaaa 

3125 

a   7 

78125 

«  9 

195312s 

a" 

48828125 

a1' 

1220703 125 

a:s 

30517578125 

I  7 

a 

7*2930453125 

a'3 

19073486328 125 

1  1 

••    476837178203 125 

11920928955078125 

298023223876953 125 

74505805969238281 

j 862645 1 4923 09570 

4656612873077392 

1 1641 53218269343 

291038304567336 

Differentia  2  " 

72759576141834 

- — 

1 8 18989403 5458 

V  nit  as—  1 

4547473 50^864 

Numerus "> 
Propo(itusJ  3 

>  a        aaa   1136868377216 

=7X1=^   284217094304 

- — 

71054273576 

Summa   — 4 

17763568394 

4440892099 

i 1 10223025 

27755575* 

69333936 

M 

*733348S 

4333371 

1083343 

270836 

67709 

16927 

4232 

ios8 

> ..  ..—,,.  — — — - 

264 

Thefe 


(4i  ) 

There  Matters  are  Quotes  from  thofe  on  the  oppofite  fide,  tbofe  being 
ZZVxlLl^  Lch  added  mkektlfibe  Log.  of  n^Place, 

5000000000000000000000000 

416666666666666666666666  I 
6z5  o 

1 1 1607142957142857 14105 

2170138888888888888888 

443892045454545454545 

9390024038461 538461 5 

20345052083333333333 

448787913602941 176 5- 
100386770148026315^ 

2270653 1 343°o59523. 
5 1830 125891644022 

11920928855078125 

2759474295 1 5^975 

642291430759295 

150213318486367 

32247067220283 

83 1 5389130495 
196647503086 1 

466407539294 
110913988021 

26438799470 

6315935429 

151179305s 

362521804 

87076316 

20947604 

S 046469 

121638s 

293788  '% 

,71039 

17196 

4167 

,  1010 

245 

59 

'5 

. 3 

549306144334054845^97^^ 
10986 12288668 10969 13 95 245 2  we 


I 

IK 
V' 
VII 
IX 
XI 
XIII 
XV 
XVVII 
XIX 
XXI 
XXIII 
XXV 
XXVII 
XXIX 
XXXI 
XXXIII 
XXXV 
XXXVII 
XXXIX 
XLI 
XLIII 
XLV 
XLVII 
XLIX 
LI 
LIU 
LV 
LVII 
LIX 
LXI 
LXIII 
LXV 
LXVII 
LXIX 
LXXI 
LXXIII 
LXXV 


Half "the  Log.  of '3. 
The  Logarithmofs. 


(  4* ; 

We  (ball  now  proceed  to  make  the  Logarithm  for  3  the  fecond 
way,  which  is  from  the  Logarithms  of  2  and  '. 

The  Infinite  Series  or  Nmbers  continually  Proportional.  Theft 
Numbers  are  continually  multiplied  by  4,  in  order  to  make  the 
Half  of  the  Logarithm  of  1 1  to  25  Places. 


a 

444 

zoooooooooooooooooooooooo 
8 

aaaaa 

32 

a  7 

128 

a  9 

St2 

a1  r 

2O48 
8l92 

Differentia      \ ' 

s 

32768 

/ 

13 IO72 

Vnitas  —  1     / 

zf 

524288 

PropofitnsS    Tf 

■=;x; 

2097152 
~~  iT                            8388608 

33554432 

Summa         2  ~~ 

) 

1342177 

53^7 

2147 

8S 

/ 


To  make  this  Infinite  Series,  Ifhould  divide  by  25  continu- 
ally, but  if  you  multiply  by  4,  and  transfer  it  anfwerable,  it 
will  be  the  fame  thing:  Becaufe^of  10  is  t*,  and  that  multi- 
plied in  it  felf ;  is  1 0t .  Therefore  multiplying  by  4  anddivi- 
£ngby  100,  is  the  fame  thing  as  multiplying  by  25  j  And 
thus  this  Infinite  Series  is  made  very  fpeedily,  in  order  to  make 
the  half  of  the  Logarithm  for  1  and  ~ . 


Tbefe 


(4?  ) 


The fe  Numbers  are  Qmtes  from  tbofe  on  the  oppofite  fide%  they  be- 
ing divided  by  1 , 3 ,  5 ,  7, 9,  1 1 ,  &C.  and  added^  make  half  the 
Logarithm  for  1  \  to  25  Places. 


I 

2000000000000000000000000 

III 

26666666666666666666666 

V 

64 

VII 

1 82857 142857 T4285714 

IX 

568888888888888888 

XI 

18618181818181818 

XIII 

6301 538461 53846 

xv 

21845333333334 

XVII 

77101 1764706 

XIX 

27594105204 

XXI 

5198643809 

XXIII 

36472209 

XXV 

1344*77 

XXVII 

49711 

XXIX 

1851 

XXXI 

6 9 

XXXIII 

3 

Half  the  Logar.  of  i  \. 

2027325540540821909890065 

The  Logarithm  of   i  -* . 

405465 108 1 08 16438 1 97801 3  I 

The  Logarithm  for  2. 

693i47i8o5599453094I7232i 

The  Logarithm  for  3 . 

1098612288668109691395245  2 

Having  now  made  the  Logarithm  for  i  '  ,  you  add  to  it  the 
Logarithm  of  2,  and  that  makes  the  Logarithm  for  3,  wl%h 
will  be  found  as  before  to  be  the  fame  Number. 

And  now  we  proceed  to  make  the  Logarithm  for  7,  and  then 
we  (hall  have  all  to  1 1 .  In  order  thereunto,  we  make  the  Lo- 
garithm for  i,|  or  i,,?  ,  and  add  that  to  the  Logarithm  of  5, 
and  it  will  produce  the  Logarithm  of  7,  for  that  1]  multiply 
by  5,  maketh7>  or  ,?X,s  =7- 


n 


m 


(  44  ) 

The  Infinite  Scries  or  Numbers  continually  Proportional :  Thefe  Num- 
bers are  continually  divided  by  36,  in  order  to  make  half  the  Lo- 
garithm of  1  3.  or  I  ,|. 

4  I 6666666666666 66666666666 

277777777777777777777777 

45296296296296295295295' 

77 1 60493  8271 60493  82716 

1 286008230452674897 1 19 

2i43347°5°754458i6i86 

3  572245084590763603 1 

5953741 80765 1272672 

9922903012752121 12 

1653 8 1 7168792020 19 

27563619479867003 

459393<5579P77864 


aaa 


A-aaaA 


Differentia 

"Unit  as 
Numerus "") 
PropofitusS 

Summa 


aaa 


C< 


76$6<$6o96662$>72 

127609349443829 

21268224907304 

354470415 12 17 

590784025203 

98464004200 

1 64 1 06673 66 

27351H228 

455851871 

75975312 

12662552 

21 IO425 

35*737 
58623 

9737 

1623 

270 


This  Series  is  made  by  dividing  Twice  by  6,  which  is  all  one 
as  if  you  divided  at  once  by  36,and  fo  every  other  Number  is  the 
proper  Number  of  the  Series  to  be  divided  by  1,3,  5,  7, 9,  1 1, 
&c.  as  in  the  other  fide  to  make  the  half  Logarithm  for  t\£* 

^F*  The  Logarithm  of  5  in  page  3  8  being  put  in  the  room 
of  that  in  page  45  will  produce  19459101490553 13305 1053  53 
for  the  true  Logarithm  of  7.  Thofe  two  laft  Numbers  in  page 
45  being  part  miftaken.  Theft 


(45) 

Thefc  Numbers  are  Quotes  from  thofe  on  the  oppofite  fide 
thofe  being  divided  by  .1.3.  5.7.  9.  11,  &c.  And  added  make 
half  the  Logarithm  of  1  \  or  1  ?„. 

*  •  •  • 

1666666666666666666666666 


I 

III 
V 

VII 

IX 

XI 

xm 

XV 

XVII 

XIX 

XXI 

XXIII 

XXV 

XXVII 

XXIX 


Half  the  Logarithm  of  1  ?. 
The  Logarithm  of  1  4  or  1 
The  Logarithm  of  5. 
The  Logarithm  of  7. 


1 543  209875543  209875543  2 
2S720N546090534979424 
5  r  03  207253  70 1 090853 
110254477819458014 
2505783589078819 
58895522820229 
14  *78  8 1550487 
34752001483 
853719335 
21707237 

550545 

14059 

35o 


1582351 1 83 105054552522972 
0   33647223^212129305045944. 

1509437912  4341003 158895254 

I9459i°I4P05  53 132473941 198 

G  Having 


(4*  ) 

Having  by  this  Calculation  made  the  half  Logarithm  of  r£# 
if  we  double  it,  and  to  that  add  the  Logarithm  of  5  that  Addi- 
tion will  produce  the  Logarithm  of  7  as  was  required.  And 
now  we  have  all  the  Logarithms  to  1 1 ,  and  to  make  the  Loga- 
rithms from  10  to  100,  it  will  not  be  much  difficult  to  proceed 
after  the  foregoing  Methods,  as  to  make  the  Logarithm  of  1 1, 
you  have  for  the  firft  Term  a->  the  refult  or  fraction  -S,  and  for 
a  a  a,  it  will  be  *£,  which  is  very  eafie  to  work.  And  for  the  Lo- 
garithm of  13  >you  make  it  of  12  multiplyed  by  1  Jy.  And  fo  it 
is  for  the  firft  Term  <*,  the  refult  or  fra&ion  1,,  and  for  the 
fecond  aaa^it  is  6  [  ,,  which  625  is=to  t>  and  fo  may  you  make 
many  eafie  compendiums  for  the  Prime  Numbers  between  10 
and  ioo,andalfonot  with  great  difficulty  from  1000  to  ioooo, 
and  when  you  have  madefomeLogarithms  you  will  perceive  how 
the  differences  arife,  and  having  for  Compofites,  Logarithms 
in  a  readinefs,  greater  and  lefler  than  the  Prime  or  Incompofite 
very  near,  it  will  be  by  the  Difference  no  great  difficulty  to 
make  a  Logarithm  for  fuch  a  Prime  very  readily  and  eafily.  And 
they  that  are  curious  herein  may  have  Compendiums  hereof  in 
James  Gregorfs  aforefaid  vera  Circuli  &  Hyperbola  Qnadratura 
to  make  Logarithms  for  Prime  or  Incompofite  Numbers,  to 
which  I  (hall  refer  him  j  and  here  I  fhall  content  my  felf  to  have 
exemplified  James  Gregory's  Method  in  his  ExercitationesGeome- 
tricdtxo  fomany  Examples  of  Logarithms  as  I  have  herein  Cal- 
culated to  25  Places,  and  fhall  in  the  next  Chapter  fhew  how  to 
produce  from  thefe  Geometrical  Logarithms  Prig£$  Loga- 
rithms. 


CHAP. 


(47) 


CHAP.    VIII. 

HAving  in  the  Preceeding  Chapter  made  the  Lqgarithnis 
for  2,3,4,  5,6,7, 8, 9,  and  10,  according  to  the  Geo- 
metrical Figure  or  Hyperbola,  I  require  the  Logarithm  of  2 
according  to  Briggs  Table.  For  as  much  as  all  Logarithms  are 
Proportional,  it  is  as  the  Hyperbolical  Logarithm  of  10,  is  to 
its  Logarithm  of  2 : :  So  is  Briggs  Logarithm  of  10  to  his  Lo- 
garithm of  2.  The  Operation  folio  wet  h, 

This  Divifor  is  half  the  Logarithm  of  10,  according  to  the 
Hyperbola, 


Q  x  ;  Divifor 


(4«  ) 

Divifor 
•  *  •  •  •  m 

i   i  j  i  2  9  2  5  4  5  4  9  7  o  2  2  8  i 

Quotient. 

301029995^3981190 

This  Quotient  is  the  Logarithm  of  2,  according  to  Briggs 
his  Table. 

By  this  Divifion  it  doth  appear,  that  this  Quotient  doth  a- 
gree  with  Briggs  his  Table  of  Logarithms  for  his  Logarithm 
Number  of  2,  whereby  it  is  apparent  he  did  produce  the  Loga- 
rithm for  2  to  15  places  very  true,  though  I  have  been  told  it 
was  eight  Perfons  work  for  a  years  time  after  his  Method, 
which  was  by  large  and  many  Extractions  of  the  Square  Root, 
and  if  it  was  fo  to  1 5  places,  it  would  have  been  very  tedious 
(ifnotimpoffible)  for  them  to  have  produced  the  Logarithm  of 
2  to  25  places,  as  before  herein  is  fhewn  and  done  by  us,  and 
both  the  Hyperbolical  and  Brig£%  Logarithm  to  25  places  may 
very  well  be  calculated  and  done  according  to  the  foregoing 
Method  in  half  a  days  time,  by  which  Method  herein  be- 
fore going  one  may  make  a  Table  of  Logarithms  in  a  fhort  Space 
to  what  Par  diem  his  Elements  of  Geometry  (a  French  Author) 
hath  declared,  for  he  faith,  he  knew  more  than  20  perfons  en- 
gaged for  20  years  with  indefatigable  afliduity  to  calculate  the 
Logarithms.  He  doth  not  fay  to  how  many  places:  But  the  great- 
eft  Radius  that  I  have  feen  of  any  French  Author  is  but  1 1  pla- 
ces, which  1  fuppofe  muft  be  but  the  fame  as  Fulactfs.  And  the 
Logarithm  for  2,  3, 4,  or  5,  &c.  to  1 1  places  according  to  the 
Method  in  this  Book  may  be  very  well  done  and  performed  in 
lefs  than  two  hours  time. 

This  Dividend  is  compounded  of  half  the  Hyperbolical  Lo- 
garithm of  2,  and  Briggs  his  Logarithm  of  10. 

Dividend 


(49  ) 

Dividend 


34^573  $90279972^4,  ©0000000000000  000 
1 185820330865797  *  - 
34^378436877419  • 
1 15079334388337338 
114630052036052851 
110137228513207981 
6520899328475928 1 
76443659599081405 
73561068092600364 
45835 153027789954 
1 1 29637663  28792697 

93 4743 7 x 4406054 3 1 
1 3  70967720843  6262 

21967517434^0339 
<»  1 044459 1 909690 1 1  op 

8295905121690561 

The  Reader  may  now  fee  that  Logarithms  derived  from  this 
Figure  or  the  Hyperbola  are  not  only  more  perceptible  and  in- 
telligible, but  with  far  more  Certainty  and  Expedition  produ- 
ced than  what  was  known  in  former  times. 

The  Divifor  in  the  foregoing  work  differs  2  Unites  in  the 
i8place,  from  the  half  Logarithm  of  1©  before  herein  calcula- 
ted, and  the  Reafon  is,  that  I  took  Gregory's  Logarithm  of  10 
in  his  Vera  Circuit  &  Hyperbola  Quadratura  de  bene  ejfe,  and  ha- 
ving calculated  the  half  Logarithm  of  2  as  before,  I  was  very 
defirous  to  fee  if  we  could  produce  Briggs  Logarithm  of  2  to 
15  places,  as  by  the  Divilionis  manifeftrand  this  I  did,  Ion q.  be- 
fore I  met  with  Gregory  %  other  Book  of  Wis  Exercitationes*& eo- 
mtrica  ;  for  fince  1  got  that  Book  I  did  calculate  De  Novo  the 
Logarithm  of  10  to  25  places  according  to  his  Doftrin  in  that 
Book,  and  as  before  herein  is  done.  And  the  Calculation  of  the 
Logarithm  of  10  as  before  doth  agree  with  Gregorfs  former 
Book  but  to  17  places ;  howfoever  the  Divifton  before  going  is 
fufficient  to  produce  Briggs  Logarithm  for  2  to  1 5  places,  and 
if  any  (hall  be  fo  curious  to  produce  Briggs  Logarithm  for  25  . 

places, 


C  5°) 

places,  he  may  rely  on  the  foregoing  Examples  herein,  and  may 
in  4  hours  time  examine  the  foregoing  Calculations  thereof,and 
in  as  little  time  produce  Brings  Logarithm  for  them  to  the  like 
Number  of  places. 

Having  this  Divifion  ready  done,  long  before  the  publifhing 
hereof,  I  have  contented  my  felf  to  iniert  it  here,  whereby  the 
ftudious  may  foon  perceive  what  to  do  further  to  gratify  him- 
felf  herein. 

I  do  not  add  hereto  any  Table  of  Logarithms,  that  being  not 
my  deflgn  at  this  time,butonly  to  (hew  how  Brigg^s  or any  other 
Logarithms  may  be  derived  from  the  Doftrin  before  going, 
and  alfo  for  the  curious  at  his  will  and  pleafure  to  examine  whe- 
ther any  Logarithms  formerly  publiftid  be  truly  made  or  not.' 

As  for  the  various  llfes  of  Logarithms  I  add  none  here,  but 
refer  the  Reader  to  fuch  Authors  (whereof  there  is  plenty,)  who 
have  long  before  written  largely  and  learnedly,  as  the  firft  In- 
ventor the  famous  Lord  Neper^  Henry  Briggs,  Edm.Gunter^Rich. 
Norwood \Wingate,and  divers  others;  as  alfo  my  Father  John  Spei- 
dellfxx  which  the  Reader  may  meet  with  many  excellent  Ufes  of 
the  Logarithms  in  all  parts  of  the  Mathematicks ;  and  I  do  find 
my  Father  printed  feveral  forts  of  Logarithms,  but  at  laft  con- 
cluded that  the  Decimal  or  Brigg%  Logarithms  were  the  belt 
fort  for  a  ftandard  Logarithm,  and  did  alfo  print  the  fame  fe- 
veral ways,  fo  ordered,  whereby  they  might  be  applyed  to  A~ 
rithmctical  Qiieftions  and  other  Operations  for  the  Solution 
thereof  with  eafe  and  readinefs. 


F    I    N    I    S. 

~    '      '        IM-V  ■■  ■  II  II— —Mill      11    ■  Jl  HI    »  ■       I— — ^— — — <l»     -  ' 


/ 


ERRATA. 

PAge  9.  Line  5,  read  7  and  8  of  Euclid. 
Page  3 1 .  Line  25.  for  ~T  read  ?£. 
Page 38 Line  10.  fori  25. read ?,44* 
Page  39.  Line  5.  for  2  -j-  i|.  read  2  *  1